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Theorem caofrss 7702
Description: Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofcom.3 (𝜑𝐺:𝐴𝑆)
caofrss.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))
Assertion
Ref Expression
caofrss (𝜑 → (𝐹r 𝑅𝐺𝐹r 𝑇𝐺))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem caofrss
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
21ffvelcdmda 7079 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelcdmda 7079 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofrss.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))
65ralrimivva 3194 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
76adantr 480 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
8 breq1 5144 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
9 breq1 5144 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑇𝑦 ↔ (𝐹𝑤)𝑇𝑦))
108, 9imbi12d 344 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑥𝑇𝑦) ↔ ((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦)))
11 breq2 5145 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
12 breq2 5145 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑇𝑦 ↔ (𝐹𝑤)𝑇(𝐺𝑤)))
1311, 12imbi12d 344 . . . . 5 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤))))
1410, 13rspc2va 3618 . . . 4 ((((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦)) → ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤)))
152, 4, 7, 14syl21anc 835 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤)))
1615ralimdva 3161 . 2 (𝜑 → (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) → ∀𝑤𝐴 (𝐹𝑤)𝑇(𝐺𝑤)))
171ffnd 6711 . . 3 (𝜑𝐹 Fn 𝐴)
183ffnd 6711 . . 3 (𝜑𝐺 Fn 𝐴)
19 caofref.1 . . 3 (𝜑𝐴𝑉)
20 inidm 4213 . . 3 (𝐴𝐴) = 𝐴
21 eqidd 2727 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
22 eqidd 2727 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
2317, 18, 19, 19, 20, 21, 22ofrfval 7676 . 2 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
2417, 18, 19, 19, 20, 21, 22ofrfval 7676 . 2 (𝜑 → (𝐹r 𝑇𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑇(𝐺𝑤)))
2516, 23, 243imtr4d 294 1 (𝜑 → (𝐹r 𝑅𝐺𝐹r 𝑇𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055   class class class wbr 5141  wf 6532  cfv 6536  r cofr 7665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ofr 7667
This theorem is referenced by: (None)
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