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Theorem fpwrelmap 28896
 Description: Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 9153 and marypha2lem1 8224. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Hypotheses
Ref Expression
fpwrelmap.1 𝐴 ∈ V
fpwrelmap.2 𝐵 ∈ V
fpwrelmap.3 𝑀 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
Assertion
Ref Expression
fpwrelmap 𝑀:(𝒫 𝐵𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem fpwrelmap
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fpwrelmap.3 . . 3 𝑀 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
2 fpwrelmap.1 . . . . . 6 𝐴 ∈ V
32a1i 11 . . . . 5 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝐴 ∈ V)
4 simpr 476 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦 ∈ (𝑓𝑥))
5 elmapi 7765 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
65ffvelrnda 6267 . . . . . . . . . 10 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝒫 𝐵)
76adantr 480 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑓𝑥) ∈ 𝒫 𝐵)
8 elelpwi 4119 . . . . . . . . 9 ((𝑦 ∈ (𝑓𝑥) ∧ (𝑓𝑥) ∈ 𝒫 𝐵) → 𝑦𝐵)
94, 7, 8syl2anc 691 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
109ex 449 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
1110alrimiv 1842 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
12 abss 3634 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
13 fpwrelmap.2 . . . . . . . 8 𝐵 ∈ V
1413ssex 4730 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1512, 14sylbir 224 . . . . . 6 (∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1611, 15syl 17 . . . . 5 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
173, 16opabex3d 7037 . . . 4 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
1817adantl 481 . . 3 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
192mptex 6390 . . . 4 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V
2019a1i 11 . . 3 ((⊤ ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V)
2110imdistanda 725 . . . . . . . . . 10 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → (𝑥𝐴𝑦𝐵)))
2221ssopab2dv 4929 . . . . . . . . 9 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2322adantr 480 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
24 simpr 476 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
25 df-xp 5044 . . . . . . . . 9 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
2625a1i 11 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2723, 24, 263sstr4d 3611 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵))
28 selpw 4115 . . . . . . 7 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵))
2927, 28sylibr 223 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
305feqmptd 6159 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
3130adantr 480 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
32 nfv 1830 . . . . . . . . 9 𝑥 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)
33 nfopab1 4651 . . . . . . . . . 10 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3433nfeq2 2766 . . . . . . . . 9 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3532, 34nfan 1816 . . . . . . . 8 𝑥(𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
36 df-rab 2905 . . . . . . . . . 10 {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)}
3736a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
38 nfv 1830 . . . . . . . . . . . 12 𝑦 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)
39 nfopab2 4652 . . . . . . . . . . . . 13 𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4039nfeq2 2766 . . . . . . . . . . . 12 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4138, 40nfan 1816 . . . . . . . . . . 11 𝑦(𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
42 nfv 1830 . . . . . . . . . . 11 𝑦 𝑥𝐴
4341, 42nfan 1816 . . . . . . . . . 10 𝑦((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴)
449adantllr 751 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
45 df-br 4584 . . . . . . . . . . . . . . . . 17 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
46 eleq2 2677 . . . . . . . . . . . . . . . . . 18 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
47 opabid 4907 . . . . . . . . . . . . . . . . . 18 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
4846, 47syl6bb 275 . . . . . . . . . . . . . . . . 17 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
4945, 48syl5bb 271 . . . . . . . . . . . . . . . 16 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5049ad2antlr 759 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
51 elfvdm 6130 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑓𝑥) → 𝑥 ∈ dom 𝑓)
5251adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥 ∈ dom 𝑓)
53 fdm 5964 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:𝐴⟶𝒫 𝐵 → dom 𝑓 = 𝐴)
545, 53syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → dom 𝑓 = 𝐴)
5554adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → dom 𝑓 = 𝐴)
5652, 55eleqtrd 2690 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝐴)
5756ex 449 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑥𝐴))
5857pm4.71rd 665 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5958ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
6050, 59bitr4d 270 . . . . . . . . . . . . . 14 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6160biimpar 501 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
6244, 61jca 553 . . . . . . . . . . . 12 ((((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑦𝐵𝑥𝑟𝑦))
6362ex 449 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → (𝑦𝐵𝑥𝑟𝑦)))
6460biimpd 218 . . . . . . . . . . . 12 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6564adantld 482 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → ((𝑦𝐵𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥)))
6663, 65impbid 201 . . . . . . . . . 10 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
6743, 66abbid 2727 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
68 abid2 2732 . . . . . . . . . 10 {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥)
6968a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥))
7037, 67, 693eqtr2rd 2651 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
7135, 70mpteq2da 4671 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7231, 71eqtrd 2644 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7329, 72jca 553 . . . . 5 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
74 ssrab2 3650 . . . . . . . . . . . 12 {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵
7513elpw2 4755 . . . . . . . . . . . 12 ({𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵 ↔ {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵)
7674, 75mpbir 220 . . . . . . . . . . 11 {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵
7776a1i 11 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
78 eqid 2610 . . . . . . . . . 10 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
7977, 78fmptd 6292 . . . . . . . . 9 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
8079adantr 480 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
81 simpr 476 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
8281feq1d 5943 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
8380, 82mpbird 246 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵)
8413pwex 4774 . . . . . . . 8 𝒫 𝐵 ∈ V
8584, 2elmap 7772 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
8683, 85sylibr 223 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
87 elpwi 4117 . . . . . . . . . 10 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵))
8887adantr 480 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
89 xpss 5149 . . . . . . . . 9 (𝐴 × 𝐵) ⊆ (V × V)
9088, 89syl6ss 3580 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
91 df-rel 5045 . . . . . . . 8 (Rel 𝑟𝑟 ⊆ (V × V))
9290, 91sylibr 223 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel 𝑟)
93 relopab 5169 . . . . . . . 8 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
9493a1i 11 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
95 id 22 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
96 nfv 1830 . . . . . . . . 9 𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
97 nfmpt1 4675 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9897nfeq2 2766 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9996, 98nfan 1816 . . . . . . . 8 𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
100 nfv 1830 . . . . . . . . 9 𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
10142nfci 2741 . . . . . . . . . . 11 𝑦𝐴
102 nfrab1 3099 . . . . . . . . . . 11 𝑦{𝑦𝐵𝑥𝑟𝑦}
103101, 102nfmpt 4674 . . . . . . . . . 10 𝑦(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
104103nfeq2 2766 . . . . . . . . 9 𝑦 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
105100, 104nfan 1816 . . . . . . . 8 𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
106 nfcv 2751 . . . . . . . 8 𝑥𝑟
107 nfcv 2751 . . . . . . . 8 𝑦𝑟
108 brelg 28801 . . . . . . . . . . . . . . . 16 ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
10987, 108sylan 487 . . . . . . . . . . . . . . 15 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
110109adantlr 747 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
111110simpld 474 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝐴)
112110simprd 478 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦𝐵)
113 simpr 476 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦)
11481fveq1d 6105 . . . . . . . . . . . . . . . . . 18 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓𝑥) = ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥))
11513rabex 4740 . . . . . . . . . . . . . . . . . . 19 {𝑦𝐵𝑥𝑟𝑦} ∈ V
11678fvmpt2 6200 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴 ∧ {𝑦𝐵𝑥𝑟𝑦} ∈ V) → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
117115, 116mpan2 703 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
118114, 117sylan9eq 2664 . . . . . . . . . . . . . . . . 17 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
119118eleq2d 2673 . . . . . . . . . . . . . . . 16 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦}))
120 rabid 3095 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦} ↔ (𝑦𝐵𝑥𝑟𝑦))
121119, 120syl6bb 275 . . . . . . . . . . . . . . 15 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
122111, 121syldan 486 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
123112, 113, 122mpbir2and 959 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥))
124111, 123jca 553 . . . . . . . . . . . 12 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
125124ex 449 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
126121simplbda 652 . . . . . . . . . . . 12 ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
127126expl 646 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦))
128125, 127impbid 201 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
12945, 128syl5bbr 273 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
130129, 47syl6bbr 277 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
13199, 105, 106, 107, 33, 39, 130eqrelrd2 28806 . . . . . . 7 (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13292, 94, 95, 131syl21anc 1317 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13386, 132jca 553 . . . . 5 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
13473, 133impbii 198 . . . 4 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
135134a1i 11 . . 3 (⊤ → ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))))
1361, 18, 20, 135f1od 6783 . 2 (⊤ → 𝑀:(𝒫 𝐵𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
137136trud 1484 1 𝑀:(𝒫 𝐵𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  Rel wrel 5043  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  fpwrelmapffs  28897
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