Theorem ovmpodf 7578

Description: Alternate deduction version of ovmpo 7582, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
ovmpodf	⊢ (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 ovmpodf

Step	Hyp	Ref	Expression
1		nfv 1946	. 2 ⊢ Ⅎ𝑥𝜑
2		ovmpodf.5	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfmpo1 7502	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
4	2, 3	nfeq 2935	. . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
5		ovmpodf.6	. . 3 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfim 1928	. 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
7		ovmpodf.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
8	7	elexd 3473	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
9		isset 3464	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	8, 9	sylib 221	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
11		nfv 1946	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
12		ovmpodf.7	. . . . 5 ⊢ Ⅎ𝑦𝐹
13		nfmpo2 7503	. . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
14	12, 13	nfeq 2935	. . . 4 ⊢ Ⅎ𝑦 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
15		ovmpodf.8	. . . 4 ⊢ Ⅎ𝑦𝜓
16	14, 15	nfim 1928	. . 3 ⊢ Ⅎ𝑦(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
17		ovmpodf.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)
18	17	elexd 3473	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
19		isset 3464	. . . 4 ⊢ (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
20	18, 19	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
21		oveq 7428	. . . . 5 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
22		simprl 783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 = 𝐴)
23		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 = 𝐵)
24	22, 23	oveq12d 7440	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
25	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐴 ∈ 𝐶)
26	22, 25	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 ∈ 𝐶)
27	17	adantrr 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐵 ∈ 𝐷)
28	23, 27	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 ∈ 𝐷)
29		ovmpodf.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)
30		eqid 2760	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
31	30	ovmpt4g 7569	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑉) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
32	26, 28, 29, 31	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
33	24, 32	eqtr3d 2797	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅)
34	33	eqeq2d 2771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) ↔ (𝐴𝐹𝐵) = 𝑅))
35		ovmpodf.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓))
36	34, 35	sylbid 243	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) → 𝜓))
37	21, 36	syl5 35	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
38	37	expr 462	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)))
39	11, 16, 20, 38	exlimimdd 2255	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
40	1, 6, 10, 39	exlimdd 2256	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907  Vcvv 3450  (class class class)co 7422   ∈ cmpo 7424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6496  df-fun 6543  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427

This theorem is used by:  ovmpodv  7579  ovmpodv2  7580