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Theorem caofrss 7658
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 7040 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelcdmda 7040 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofrss.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))
65ralrimivva 3198 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
76adantr 482 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
8 breq1 5113 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
9 breq1 5113 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑇𝑦 ↔ (𝐹𝑤)𝑇𝑦))
108, 9imbi12d 345 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑥𝑇𝑦) ↔ ((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦)))
11 breq2 5114 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
12 breq2 5114 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑇𝑦 ↔ (𝐹𝑤)𝑇(𝐺𝑤)))
1311, 12imbi12d 345 . . . . 5 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤))))
1410, 13rspc2va 3594 . . . 4 ((((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦)) → ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤)))
152, 4, 7, 14syl21anc 837 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤)))
1615ralimdva 3165 . 2 (𝜑 → (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) → ∀𝑤𝐴 (𝐹𝑤)𝑇(𝐺𝑤)))
171ffnd 6674 . . 3 (𝜑𝐹 Fn 𝐴)
183ffnd 6674 . . 3 (𝜑𝐺 Fn 𝐴)
19 caofref.1 . . 3 (𝜑𝐴𝑉)
20 inidm 4183 . . 3 (𝐴𝐴) = 𝐴
21 eqidd 2738 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
22 eqidd 2738 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
2317, 18, 19, 19, 20, 21, 22ofrfval 7632 . 2 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
2417, 18, 19, 19, 20, 21, 22ofrfval 7632 . 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 397   = wceq 1542  wcel 2107  wral 3065   class class class wbr 5110  wf 6497  cfv 6501  r cofr 7621
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ofr 7623
This theorem is referenced by: (None)
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