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Theorem ovmpt2rdxf 41910
 Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6684. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
ovmpt2rdx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpt2rdx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2rdx.3 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
ovmpt2rdx.4 (𝜑𝐴𝐿)
ovmpt2rdx.5 (𝜑𝐵𝐷)
ovmpt2rdx.6 (𝜑𝑆𝑋)
ovmpt2rdxf.px 𝑥𝜑
ovmpt2rdxf.py 𝑦𝜑
ovmpt2rdxf.ay 𝑦𝐴
ovmpt2rdxf.bx 𝑥𝐵
ovmpt2rdxf.sx 𝑥𝑆
ovmpt2rdxf.sy 𝑦𝑆
Assertion
Ref Expression
ovmpt2rdxf (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpt2rdxf
StepHypRef Expression
1 ovmpt2rdx.1 . . 3 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
21oveqd 6566 . 2 (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3 ovmpt2rdx.4 . . . 4 (𝜑𝐴𝐿)
4 ovmpt2rdxf.px . . . . 5 𝑥𝜑
5 ovmpt2rdx.5 . . . . . 6 (𝜑𝐵𝐷)
6 ovmpt2rdxf.py . . . . . . 7 𝑦𝜑
7 eqid 2610 . . . . . . . . 9 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
87ovmpt4g 6681 . . . . . . . 8 ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
98a1i 11 . . . . . . 7 (𝜑 → ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
106, 9alrimi 2069 . . . . . 6 (𝜑 → ∀𝑦((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
115, 10spsbcd 3416 . . . . 5 (𝜑[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
124, 11alrimi 2069 . . . 4 (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
133, 12spsbcd 3416 . . 3 (𝜑[𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
145adantr 480 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
153ad2antrr 758 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴𝐿)
16 simpr 476 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1716adantr 480 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
18 ovmpt2rdx.3 . . . . . . . . 9 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
1918adantlr 747 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
2015, 17, 193eltr4d 2703 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐶)
215ad2antrr 758 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵𝐷)
22 eleq1 2676 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
2322adantl 481 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑦𝐷𝐵𝐷))
2421, 23mpbird 246 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦𝐷)
25 ovmpt2rdx.2 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
2625anassrs 678 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
27 ovmpt2rdx.6 . . . . . . . . 9 (𝜑𝑆𝑋)
2827ad2antrr 758 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆𝑋)
2926, 28eqeltrd 2688 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅𝑋)
30 biimt 349 . . . . . . 7 ((𝑥𝐶𝑦𝐷𝑅𝑋) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3120, 24, 29, 30syl3anc 1318 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
32 simpr 476 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3317, 32oveq12d 6567 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3433, 26eqeq12d 2625 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
3531, 34bitr3d 269 . . . . 5 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
36 ovmpt2rdxf.ay . . . . . . 7 𝑦𝐴
3736nfeq2 2766 . . . . . 6 𝑦 𝑥 = 𝐴
386, 37nfan 1816 . . . . 5 𝑦(𝜑𝑥 = 𝐴)
39 nfmpt22 6621 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
40 nfcv 2751 . . . . . . . 8 𝑦𝐵
4136, 39, 40nfov 6575 . . . . . . 7 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
42 ovmpt2rdxf.sy . . . . . . 7 𝑦𝑆
4341, 42nfeq 2762 . . . . . 6 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
4443a1i 11 . . . . 5 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
4514, 35, 38, 44sbciedf 3438 . . . 4 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
46 nfcv 2751 . . . . . . 7 𝑥𝐴
47 nfmpt21 6620 . . . . . . 7 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
48 ovmpt2rdxf.bx . . . . . . 7 𝑥𝐵
4946, 47, 48nfov 6575 . . . . . 6 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
50 ovmpt2rdxf.sx . . . . . 6 𝑥𝑆
5149, 50nfeq 2762 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
5251a1i 11 . . . 4 (𝜑 → Ⅎ𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
533, 45, 4, 52sbciedf 3438 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
5413, 53mpbid 221 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
552, 54eqtrd 2644 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  [wsbc 3402  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  ovmpt2rdx  41911
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