Theorem rabfodom 28728

Description: Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Hypotheses

Ref	Expression
rabfodom.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
rabfodom.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rabfodom.3	⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)

Assertion

Ref	Expression
rabfodom	⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem rabfodom

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3176	. . . . . 6 ⊢ 𝑎 ∈ V
2	1	rabex 4740	. . . . 5 ⊢ {𝑥 ∈ 𝑎 ∣ 𝜓} ∈ V
3		eqid 2610	. . . . . 6 ⊢ (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥))
4		rabfodom.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)
5		fof 6028	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7	6	feqmptd 6159	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
8	7	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	8	reseq1d 5316	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ 𝑎))
10		elpwi 4117	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
11	10	ad2antlr 759	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → 𝑎 ⊆ 𝐴)
12	11	resmptd 5371	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)))
13	9, 12	eqtrd 2644	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)))
14		f1oeq1 6040	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) → ((𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵 ↔ (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵))
15	14	biimpa 500	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑎) = (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵)
16	13, 15	sylancom 698	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → (𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)):𝑎–1-1-onto→𝐵)
17		simp1ll 1117	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝜑)
18	11	3ad2ant1 1075	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑎 ⊆ 𝐴)
19		simp2 1055	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝑎)
20	18, 19	sseldd 3569	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑥 ∈ 𝐴)
21		simp3 1056	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
22		rabfodom.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
23	17, 20, 21, 22	syl3anc 1318	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝑎 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))
24	3, 16, 23	f1oresrab 6302	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → ((𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) ↾ {𝑥 ∈ 𝑎 ∣ 𝜓}):{𝑥 ∈ 𝑎 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
25		f1oeng 7860	. . . . 5 ⊢ (({𝑥 ∈ 𝑎 ∣ 𝜓} ∈ V ∧ ((𝑥 ∈ 𝑎 ↦ (𝐹‘𝑥)) ↾ {𝑥 ∈ 𝑎 ∣ 𝜓}):{𝑥 ∈ 𝑎 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≈ {𝑦 ∈ 𝐵 ∣ 𝜒})
26	2, 24, 25	sylancr 694	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≈ {𝑦 ∈ 𝐵 ∣ 𝜒})
27	26	ensymd 7893	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≈ {𝑥 ∈ 𝑎 ∣ 𝜓})
28		rabfodom.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
29		rabexg 4739	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
31	30	ad2antrr 758	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
32		rabss2 3648	. . . . 5 ⊢ (𝑎 ⊆ 𝐴 → {𝑥 ∈ 𝑎 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
33	11, 32	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
34		ssdomg 7887	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V → ({𝑥 ∈ 𝑎 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} → {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}))
35	31, 33, 34	sylc 63	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
36		endomtr 7900	. . 3 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜒} ≈ {𝑥 ∈ 𝑎 ∣ 𝜓} ∧ {𝑥 ∈ 𝑎 ∣ 𝜓} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓}) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
37	27, 35, 36	syl2anc 691	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 𝐴) ∧ (𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵) → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
38		foresf1o 28727	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
39	28, 4, 38	syl2anc 691	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝒫 𝐴(𝐹 ↾ 𝑎):𝑎–1-1-onto→𝐵)
40	37, 39	r19.29a 3060	1 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643   ↾ cres 5040  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804   ≈ cen 7838   ≼ cdom 7839

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  ax-reg 8380  ax-inf2 8421  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822

This theorem is referenced by:  locfinreflem  29235