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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 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | offval.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | fnex 6972 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 1, 2, 3 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | fnex 6972 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V) | |
8 | 5, 6, 7 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
9 | dmeq 5765 | . . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
10 | dmeq 5765 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
11 | 9, 10 | ineqan12d 4189 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺)) |
12 | fveq1 6662 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
13 | fveq1 6662 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
14 | 12, 13 | breqan12d 5073 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
15 | 11, 14 | raleqbidv 3400 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
16 | df-ofr 7402 | . . . 4 ⊢ ∘r 𝑅 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} | |
17 | 15, 16 | brabga 5412 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
18 | 4, 8, 17 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
19 | 1 | fndmd 6449 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
20 | 5 | fndmd 6449 | . . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵) |
21 | 19, 20 | ineq12d 4188 | . . . 4 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
22 | offval.5 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
23 | 21, 22 | syl6eq 2870 | . . 3 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆) |
24 | 23 | raleqdv 3414 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
25 | inss1 4203 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
26 | 22, 25 | eqsstrri 4000 | . . . . . 6 ⊢ 𝑆 ⊆ 𝐴 |
27 | 26 | sseli 3961 | . . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴) |
28 | offval.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | |
29 | 27, 28 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) = 𝐶) |
30 | inss2 4204 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
31 | 22, 30 | eqsstrri 4000 | . . . . . 6 ⊢ 𝑆 ⊆ 𝐵 |
32 | 31 | sseli 3961 | . . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐵) |
33 | offval.7 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) | |
34 | 32, 33 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) = 𝐷) |
35 | 29, 34 | breq12d 5070 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ 𝐶𝑅𝐷)) |
36 | 35 | ralbidva 3194 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) |
37 | 18, 24, 36 | 3bitrd 307 | 1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 Vcvv 3493 ∩ cin 3933 class class class wbr 5057 dom cdm 5548 Fn wfn 6343 ‘cfv 6348 ∘r cofr 7400 |
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-rep 5181 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-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 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-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ofr 7402 |
This theorem is referenced by: ofrval 7411 ofrfval2 7419 caofref 7427 caofrss 7434 caoftrn 7436 ofsubge0 11629 pwsle 16757 pwsleval 16758 psrbaglesupp 20140 psrbagcon 20143 psrbaglefi 20144 psrlidm 20175 0plef 24265 0pledm 24266 itg1ge0 24279 mbfi1fseqlem5 24312 xrge0f 24324 itg2ge0 24328 itg2lea 24337 itg2splitlem 24341 itg2monolem1 24343 itg2mono 24346 itg2i1fseqle 24347 itg2i1fseq 24348 itg2addlem 24351 itg2cnlem1 24354 itg2addnclem 34935 |
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