Theorem ovmpodv 6186

Description: Alternate deduction version of ovmpo 6189, 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 2384	. 2 ⊢ Ⅎ𝑥𝐹
6		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
7		nfcv 2384	. 2 ⊢ Ⅎ𝑦𝐹
8		nfv 1577	. 2 ⊢ Ⅎ𝑦𝜓
9	1, 2, 3, 4, 5, 6, 7, 8	ovmpodf 6185	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  (class class class)co 6050   ∈ cmpo 6052

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055

This theorem is referenced by: (None)