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Mirrors > Home > MPE Home > Th. List > ovmpoga | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ovmpoga.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
ovmpoga.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
Ref | Expression |
---|---|
ovmpoga | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3465 | . 2 ⊢ (𝑆 ∈ 𝐻 → 𝑆 ∈ V) | |
2 | ovmpoga.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
4 | ovmpoga.1 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) | |
5 | 4 | adantl 483 | . . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
6 | simp1 1137 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝐴 ∈ 𝐶) | |
7 | simp2 1138 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝐵 ∈ 𝐷) | |
8 | simp3 1139 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝑆 ∈ V) | |
9 | 3, 5, 6, 7, 8 | ovmpod 7511 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆) |
10 | 1, 9 | syl3an3 1166 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3447 (class class class)co 7361 ∈ cmpo 7363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 |
This theorem is referenced by: ovmpoa 7514 ovmpog 7518 elovmpo 7602 offval 7630 offval3 7919 mptmpoopabbrdOLD 8019 bropopvvv 8026 reps 14667 hashbcval 16882 setsvalg 17046 ressval 17123 restval 17316 sylow1lem4 19391 sylow3lem2 19418 sylow3lem3 19419 lsmvalx 19429 mvrfval 21412 opsrval 21470 marrepfval 21932 marrepval0 21933 marepvfval 21937 marepvval0 21938 cnmpt12 23041 cnmpt22 23048 qtopval 23069 flimval 23337 fclsval 23382 ucnval 23652 stdbdmetval 23893 fldgenval 32135 resvval 32172 irngval 32423 minplyval 32436 ofcfval3 32765 fmulcl 43912 |
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