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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 6971 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 1, 2, 3 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | fnex 6971 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V) | |
8 | 5, 6, 7 | syl2anc 584 | . . 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 4188 | . . . . 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 3399 | . . . 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 7399 | . . . 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 584 | . 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 4187 | . . . 4 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
22 | offval.5 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
23 | 21, 22 | syl6eq 2869 | . . 3 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆) |
24 | 23 | raleqdv 3413 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
25 | inss1 4202 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
26 | 22, 25 | eqsstrri 3999 | . . . . . 6 ⊢ 𝑆 ⊆ 𝐴 |
27 | 26 | sseli 3960 | . . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴) |
28 | offval.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | |
29 | 27, 28 | sylan2 592 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) = 𝐶) |
30 | inss2 4203 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
31 | 22, 30 | eqsstrri 3999 | . . . . . 6 ⊢ 𝑆 ⊆ 𝐵 |
32 | 31 | sseli 3960 | . . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐵) |
33 | offval.7 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) | |
34 | 32, 33 | sylan2 592 | . . . 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 3193 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) |
37 | 18, 24, 36 | 3bitrd 306 | 1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∩ cin 3932 class class class wbr 5057 dom cdm 5548 Fn wfn 6343 ‘cfv 6348 ∘r cofr 7397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 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 7399 |
This theorem is referenced by: ofrval 7408 ofrfval2 7416 caofref 7424 caofrss 7431 caoftrn 7433 ofsubge0 11625 pwsle 16753 pwsleval 16754 psrbaglesupp 20076 psrbagcon 20079 psrbaglefi 20080 psrlidm 20111 0plef 24200 0pledm 24201 itg1ge0 24214 mbfi1fseqlem5 24247 xrge0f 24259 itg2ge0 24263 itg2lea 24272 itg2splitlem 24276 itg2monolem1 24278 itg2mono 24281 itg2i1fseqle 24282 itg2i1fseq 24283 itg2addlem 24286 itg2cnlem1 24289 itg2addnclem 34824 |
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