Theorem ovmpodf 5973

Description: Alternate deduction version of ovmpo 5977, 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 1516	. 2 ⊢ Ⅎ𝑥𝜑
2		ovmpodf.5	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfmpo1 5909	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
4	2, 3	nfeq 2316	. . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
5		ovmpodf.6	. . 3 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfim 1560	. 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
7		ovmpodf.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
8		elex 2737	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
9	7, 8	syl 14	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
10		isset 2732	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
11	9, 10	sylib 121	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
12		ovmpodf.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)
13		elex 2737	. . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
14	12, 13	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
15		isset 2732	. . . 4 ⊢ (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
16	14, 15	sylib 121	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
17		nfv 1516	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
18		ovmpodf.7	. . . . . 6 ⊢ Ⅎ𝑦𝐹
19		nfmpo2 5910	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
20	18, 19	nfeq 2316	. . . . 5 ⊢ Ⅎ𝑦 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
21		ovmpodf.8	. . . . 5 ⊢ Ⅎ𝑦𝜓
22	20, 21	nfim 1560	. . . 4 ⊢ Ⅎ𝑦(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
23		oveq 5848	. . . . . 6 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
24		simprl 521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 = 𝐴)
25		simprr 522	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 = 𝐵)
26	24, 25	oveq12d 5860	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
27	7	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐴 ∈ 𝐶)
28	24, 27	eqeltrd 2243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 ∈ 𝐶)
29	12	adantrr 471	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐵 ∈ 𝐷)
30	25, 29	eqeltrd 2243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 ∈ 𝐷)
31		ovmpodf.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)
32		eqid 2165	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
33	32	ovmpt4g 5964	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑉) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
34	28, 30, 31, 33	syl3anc 1228	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
35	26, 34	eqtr3d 2200	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅)
36	35	eqeq2d 2177	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) ↔ (𝐴𝐹𝐵) = 𝑅))
37		ovmpodf.4	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓))
38	36, 37	sylbid 149	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) → 𝜓))
39	23, 38	syl5 32	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
40	39	expr 373	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)))
41	17, 22, 40	exlimd 1585	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (∃𝑦 𝑦 = 𝐵 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)))
42	16, 41	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))
43	1, 6, 11, 42	exlimdd 1860	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  Ⅎwnf 1448  ∃wex 1480   ∈ wcel 2136  Ⅎwnfc 2295  Vcvv 2726  (class class class)co 5842   ∈ cmpo 5844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847

This theorem is referenced by:  ovmpodv  5974  ovmpodv2  5975