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Theorem ovmpt2rdxf 42635
Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6932. (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 6809 . 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 2770 . . . . . . . . 9 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
87ovmpt4g 6929 . . . . . . . 8 ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
98a1i 11 . . . . . . 7 (𝜑 → ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
106, 9alrimi 2237 . . . . . 6 (𝜑 → ∀𝑦((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
115, 10spsbcd 3599 . . . . 5 (𝜑[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
124, 11alrimi 2237 . . . 4 (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
133, 12spsbcd 3599 . . 3 (𝜑[𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
145adantr 466 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
153ad2antrr 697 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴𝐿)
16 simpr 471 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1716adantr 466 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
18 ovmpt2rdx.3 . . . . . . . . 9 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
1918adantlr 686 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
2015, 17, 193eltr4d 2864 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐶)
215ad2antrr 697 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵𝐷)
22 eleq1 2837 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
2322adantl 467 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑦𝐷𝐵𝐷))
2421, 23mpbird 247 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦𝐷)
25 ovmpt2rdx.2 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
2625anassrs 458 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
27 ovmpt2rdx.6 . . . . . . . . 9 (𝜑𝑆𝑋)
2827ad2antrr 697 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆𝑋)
2926, 28eqeltrd 2849 . . . . . . 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 1475 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
32 simpr 471 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3317, 32oveq12d 6810 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3433, 26eqeq12d 2785 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
3531, 34bitr3d 270 . . . . 5 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
36 ovmpt2rdxf.ay . . . . . . 7 𝑦𝐴
3736nfeq2 2928 . . . . . 6 𝑦 𝑥 = 𝐴
386, 37nfan 1979 . . . . 5 𝑦(𝜑𝑥 = 𝐴)
39 nfmpt22 6869 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
40 nfcv 2912 . . . . . . . 8 𝑦𝐵
4136, 39, 40nfov 6820 . . . . . . 7 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
42 ovmpt2rdxf.sy . . . . . . 7 𝑦𝑆
4341, 42nfeq 2924 . . . . . 6 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
4443a1i 11 . . . . 5 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
4514, 35, 38, 44sbciedf 3621 . . . 4 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
46 nfcv 2912 . . . . . . 7 𝑥𝐴
47 nfmpt21 6868 . . . . . . 7 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
48 ovmpt2rdxf.bx . . . . . . 7 𝑥𝐵
4946, 47, 48nfov 6820 . . . . . 6 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
50 ovmpt2rdxf.sx . . . . . 6 𝑥𝑆
5149, 50nfeq 2924 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
5251a1i 11 . . . 4 (𝜑 → Ⅎ𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
533, 45, 4, 52sbciedf 3621 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
5413, 53mpbid 222 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
552, 54eqtrd 2804 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wnf 1855  wcel 2144  wnfc 2899  [wsbc 3585  (class class class)co 6792  cmpt2 6794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797
This theorem is referenced by:  ovmpt2rdx  42636
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