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Mirrors > Home > MPE Home > Th. List > ofrfval | Structured version Visualization version GIF version |
Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
offval.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
offval.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
offval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offval.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
offval.5 | ⊢ (𝐴 ∩ 𝐵) = 𝑆 |
offval.6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) |
offval.7 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) |
Ref | Expression |
---|---|
ofrfval | ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | offval.2 | . 2 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
3 | offval.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | 1, 3 | fnexd 6991 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | offval.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
6 | 2, 5 | fnexd 6991 | . 2 ⊢ (𝜑 → 𝐺 ∈ V) |
7 | offval.5 | . 2 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
8 | offval.6 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | |
9 | offval.7 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) | |
10 | 1, 2, 4, 6, 7, 8, 9 | ofrfvalg 7432 | 1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 Vcvv 3398 ∩ cin 3842 class class class wbr 5030 Fn wfn 6334 ‘cfv 6339 ∘r cofr 7424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ofr 7426 |
This theorem is referenced by: ofrval 7436 ofrfval2 7445 caofref 7453 caofrss 7460 caoftrn 7462 ofsubge0 11715 psrbaglesuppOLD 20738 psrbagcon 20743 psrbagconOLD 20744 psrbaglefiOLD 20746 psrlidm 20782 0plef 24424 0pledm 24425 itg1ge0 24438 mbfi1fseqlem5 24472 xrge0f 24484 itg2ge0 24488 itg2lea 24497 itg2splitlem 24501 itg2monolem1 24503 itg2mono 24506 itg2i1fseqle 24507 itg2i1fseq 24508 itg2addlem 24511 itg2cnlem1 24514 itg2addnclem 35451 |
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