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Theorem ovmpordxf 44378
Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7292. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
ovmpordx.1 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
ovmpordx.2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpordx.3 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
ovmpordx.4 (𝜑𝐴𝐿)
ovmpordx.5 (𝜑𝐵𝐷)
ovmpordx.6 (𝜑𝑆𝑋)
ovmpordxf.px 𝑥𝜑
ovmpordxf.py 𝑦𝜑
ovmpordxf.ay 𝑦𝐴
ovmpordxf.bx 𝑥𝐵
ovmpordxf.sx 𝑥𝑆
ovmpordxf.sy 𝑦𝑆
Assertion
Ref Expression
ovmpordxf (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 ovmpordxf
StepHypRef Expression
1 ovmpordx.1 . . 3 (𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
21oveqd 7165 . 2 (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3 ovmpordx.4 . . . 4 (𝜑𝐴𝐿)
4 ovmpordxf.px . . . . 5 𝑥𝜑
5 ovmpordx.5 . . . . . 6 (𝜑𝐵𝐷)
6 ovmpordxf.py . . . . . . 7 𝑦𝜑
7 eqid 2819 . . . . . . . . 9 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
87ovmpt4g 7289 . . . . . . . 8 ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
98a1i 11 . . . . . . 7 (𝜑 → ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
106, 9alrimi 2206 . . . . . 6 (𝜑 → ∀𝑦((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
115, 10spsbcd 3784 . . . . 5 (𝜑[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
124, 11alrimi 2206 . . . 4 (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
133, 12spsbcd 3784 . . 3 (𝜑[𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅))
145adantr 483 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
153ad2antrr 724 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴𝐿)
16 simpr 487 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1716adantr 483 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
18 ovmpordx.3 . . . . . . . . 9 ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)
1918adantlr 713 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
2015, 17, 193eltr4d 2926 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐶)
215ad2antrr 724 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵𝐷)
22 eleq1 2898 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
2322adantl 484 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑦𝐷𝐵𝐷))
2421, 23mpbird 259 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦𝐷)
25 ovmpordx.2 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
2625anassrs 470 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
27 ovmpordx.6 . . . . . . . . 9 (𝜑𝑆𝑋)
2827ad2antrr 724 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆𝑋)
2926, 28eqeltrd 2911 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅𝑋)
30 biimt 363 . . . . . . 7 ((𝑥𝐶𝑦𝐷𝑅𝑋) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
3120, 24, 29, 30syl3anc 1366 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ ((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)))
32 simpr 487 . . . . . . . 8 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3317, 32oveq12d 7166 . . . . . . 7 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
3433, 26eqeq12d 2835 . . . . . 6 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
3531, 34bitr3d 283 . . . . 5 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
36 ovmpordxf.ay . . . . . . 7 𝑦𝐴
3736nfeq2 2993 . . . . . 6 𝑦 𝑥 = 𝐴
386, 37nfan 1894 . . . . 5 𝑦(𝜑𝑥 = 𝐴)
39 nfmpo2 7227 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
40 nfcv 2975 . . . . . . . 8 𝑦𝐵
4136, 39, 40nfov 7178 . . . . . . 7 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
42 ovmpordxf.sy . . . . . . 7 𝑦𝑆
4341, 42nfeq 2989 . . . . . 6 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
4443a1i 11 . . . . 5 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
4514, 35, 38, 44sbciedf 3811 . . . 4 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
46 nfcv 2975 . . . . . . 7 𝑥𝐴
47 nfmpo1 7226 . . . . . . 7 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
48 ovmpordxf.bx . . . . . . 7 𝑥𝐵
4946, 47, 48nfov 7178 . . . . . 6 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
50 ovmpordxf.sx . . . . . 6 𝑥𝑆
5149, 50nfeq 2989 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
5251a1i 11 . . . 4 (𝜑 → Ⅎ𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
533, 45, 4, 52sbciedf 3811 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥𝐶𝑦𝐷𝑅𝑋) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
5413, 53mpbid 234 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
552, 54eqtrd 2854 1 (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wnf 1778  wcel 2108  wnfc 2959  [wsbc 3770  (class class class)co 7148  cmpo 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153
This theorem is referenced by:  ovmpordx  44379
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