Theorem ovmpodv 6101

Description: Alternate deduction version of ovmpo 6104, 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 2350	. 2 ⊢ Ⅎ𝑥𝐹
6		nfv 1552	. 2 ⊢ Ⅎ𝑥𝜓
7		nfcv 2350	. 2 ⊢ Ⅎ𝑦𝐹
8		nfv 1552	. 2 ⊢ Ⅎ𝑦𝜓
9	1, 2, 3, 4, 5, 6, 7, 8	ovmpodf 6100	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2178  (class class class)co 5967   ∈ cmpo 5969

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 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972

This theorem is referenced by: (None)