Theorem ovmpt2rdx 43133

Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 7063. (Contributed by AV, 30-Mar-2019.)

Hypotheses

Ref	Expression
ovmpt2rdx.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
ovmpt2rdx.2	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2rdx.3	⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
ovmpt2rdx.4	⊢ (𝜑 → 𝐴 ∈ 𝐿)
ovmpt2rdx.5	⊢ (𝜑 → 𝐵 ∈ 𝐷)
ovmpt2rdx.6	⊢ (𝜑 → 𝑆 ∈ 𝑋)

Assertion

Ref	Expression
ovmpt2rdx	⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑦,𝐴   𝑥,𝐵   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpt2rdx

Step	Hyp	Ref	Expression
1		ovmpt2rdx.1	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
2		ovmpt2rdx.2	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
3		ovmpt2rdx.3	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
4		ovmpt2rdx.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐿)
5		ovmpt2rdx.5	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
6		ovmpt2rdx.6	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋)
7		nfv 1957	. 2 ⊢ Ⅎ𝑥𝜑
8		nfv 1957	. 2 ⊢ Ⅎ𝑦𝜑
9		nfcv 2934	. 2 ⊢ Ⅎ𝑦𝐴
10		nfcv 2934	. 2 ⊢ Ⅎ𝑥𝐵
11		nfcv 2934	. 2 ⊢ Ⅎ𝑥𝑆
12		nfcv 2934	. 2 ⊢ Ⅎ𝑦𝑆
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpt2rdxf 43132	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  (class class class)co 6922   ↦ cmpt2 6924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927

This theorem is referenced by:  ovmpt2x2  43134