Theorem ovmpodf 6090

Description: Alternate deduction version of ovmpo 6094, 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 1552	. 2 ⊢ Ⅎ𝑥𝜑
2		ovmpodf.5	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfmpo1 6025	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
4	2, 3	nfeq 2357	. . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
5		ovmpodf.6	. . 3 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfim 1596	. 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
7		ovmpodf.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
8		elex 2785	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
9	7, 8	syl 14	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
10		isset 2780	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
11	9, 10	sylib 122	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
12		ovmpodf.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)
13		elex 2785	. . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
14	12, 13	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
15		isset 2780	. . . 4 ⊢ (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
16	14, 15	sylib 122	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
17		nfv 1552	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
18		ovmpodf.7	. . . . . 6 ⊢ Ⅎ𝑦𝐹
19		nfmpo2 6026	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
20	18, 19	nfeq 2357	. . . . 5 ⊢ Ⅎ𝑦 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
21		ovmpodf.8	. . . . 5 ⊢ Ⅎ𝑦𝜓
22	20, 21	nfim 1596	. . . 4 ⊢ Ⅎ𝑦(𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)
23		oveq 5963	. . . . . 6 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
24		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 = 𝐴)
25		simprr 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 = 𝐵)
26	24, 25	oveq12d 5975	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
27	7	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐴 ∈ 𝐶)
28	24, 27	eqeltrd 2283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑥 ∈ 𝐶)
29	12	adantrr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐵 ∈ 𝐷)
30	25, 29	eqeltrd 2283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑦 ∈ 𝐷)
31		ovmpodf.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)
32		eqid 2206	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
33	32	ovmpt4g 6081	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑉) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
34	28, 30, 31, 33	syl3anc 1250	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
35	26, 34	eqtr3d 2241	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅)
36	35	eqeq2d 2218	. . . . . . 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 150	. . . . . 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 375	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)))
41	17, 22, 40	exlimd 1621	. . 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 1896	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  Ⅎwnf 1484  ∃wex 1516   ∈ wcel 2177  Ⅎwnfc 2336  Vcvv 2773  (class class class)co 5957   ∈ cmpo 5959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962

This theorem is referenced by:  ovmpodv  6091  ovmpodv2  6092