Theorem ovmpt2df 6690

Description: Alternate deduction version of ovmpt2 6694, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
ovmpt2df.1	⊢ (𝜑 → 𝐴 ∈ 𝐶)
ovmpt2df.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)
ovmpt2df.3	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)
ovmpt2df.4	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓))
ovmpt2df.5	⊢ Ⅎ𝑥𝐹
ovmpt2df.6	⊢ Ⅎ𝑥𝜓
ovmpt2df.7	⊢ Ⅎ𝑦𝐹
ovmpt2df.8	⊢ Ⅎ𝑦𝜓

Assertion

Ref	Expression
ovmpt2df	⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpt2df

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑥𝜑
2		ovmpt2df.5	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfmpt21 6620	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
4	2, 3	nfeq 2762	. . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
5		ovmpt2df.6	. . 3 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfim 1813	. 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
7		ovmpt2df.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
8		elex 3185	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
10		isset 3180	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
11	9, 10	sylib 207	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
12		ovmpt2df.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)
13		elex 3185	. . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
14	12, 13	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
15		isset 3180	. . . 4 ⊢ (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
16	14, 15	sylib 207	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
17		nfv 1830	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
18		ovmpt2df.7	. . . . . 6 ⊢ Ⅎ𝑦𝐹
19		nfmpt22 6621	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
20	18, 19	nfeq 2762	. . . . 5 ⊢ Ⅎ𝑦 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
21		ovmpt2df.8	. . . . 5 ⊢ Ⅎ𝑦𝜓
22	20, 21	nfim 1813	. . . 4 ⊢ Ⅎ𝑦(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
23		oveq 6555	. . . . . 6 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
24		simprl 790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 = 𝐴)
25		simprr 792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 = 𝐵)
26	24, 25	oveq12d 6567	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
27	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐴 ∈ 𝐶)
28	24, 27	eqeltrd 2688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 ∈ 𝐶)
29	12	adantrr 749	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐵 ∈ 𝐷)
30	25, 29	eqeltrd 2688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 ∈ 𝐷)
31		ovmpt2df.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)
32		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
33	32	ovmpt4g 6681	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑉) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
34	28, 30, 31, 33	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
35	26, 34	eqtr3d 2646	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅)
36	35	eqeq2d 2620	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) ↔ (𝐴𝐹𝐵) = 𝑅))
37		ovmpt2df.4	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓))
38	36, 37	sylbid 229	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) → 𝜓))
39	23, 38	syl5 33	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
40	39	expr 641	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)))
41	17, 22, 40	exlimd 2074	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (∃𝑦 𝑦 = 𝐵 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)))
42	16, 41	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
43	1, 6, 11, 42	exlimdd 2075	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173  (class class class)co 6549   ↦ cmpt2 6551

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  ovmpt2dv  6691  ovmpt2dv2  6692