Theorem ovmpodv 7421

Description: Alternate deduction version of ovmpo 7424, 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))

Assertion

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

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

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

Proof of Theorem ovmpodv

Step	Hyp	Ref	Expression
1		ovmpodf.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
2		ovmpodf.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)
3		ovmpodf.3	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉)
4		ovmpodf.4	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓))
5		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝐹
6		nfv 1920	. 2 ⊢ Ⅎ𝑥𝜓
7		nfcv 2908	. 2 ⊢ Ⅎ𝑦𝐹
8		nfv 1920	. 2 ⊢ Ⅎ𝑦𝜓
9	1, 2, 3, 4, 5, 6, 7, 8	ovmpodf 7420	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  (class class class)co 7268   ∈ cmpo 7270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273

This theorem is referenced by:  xpcco  17881  curf12  17926  curf2  17928