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 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)

Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5 (𝜑𝐹:𝐴𝑆)
21ffvelrnda 6844 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 6844 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofrss.4 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))
65ralrimivva 3189 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
76adantr 483 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑅𝑦𝑥𝑇𝑦))
8 breq1 5060 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦 ↔ (𝐹𝑤)𝑅𝑦))
9 breq1 5060 . . . . . 6 (𝑥 = (𝐹𝑤) → (𝑥𝑇𝑦 ↔ (𝐹𝑤)𝑇𝑦))
108, 9imbi12d 347 . . . . 5 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦𝑥𝑇𝑦) ↔ ((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦)))
11 breq2 5061 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦 ↔ (𝐹𝑤)𝑅(𝐺𝑤)))
12 breq2 5061 . . . . . 6 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑇𝑦 ↔ (𝐹𝑤)𝑇(𝐺𝑤)))
1311, 12imbi12d 347 . . . . 5 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦 → (𝐹𝑤)𝑇𝑦) ↔ ((𝐹𝑤)𝑅(𝐺𝑤) → (𝐹𝑤)𝑇(𝐺𝑤))))
1410, 13rspc2va 3632 . . . 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 3175 . 2 (𝜑 → (∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤) → ∀𝑤𝐴 (𝐹𝑤)𝑇(𝐺𝑤)))
171ffnd 6508 . . 3 (𝜑𝐹 Fn 𝐴)
183ffnd 6508 . . 3 (𝜑𝐺 Fn 𝐴)
19 caofref.1 . . 3 (𝜑𝐴𝑉)
20 inidm 4193 . . 3 (𝐴𝐴) = 𝐴
21 eqidd 2820 . . 3 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
22 eqidd 2820 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
2317, 18, 19, 19, 20, 21, 22ofrfval 7409 . 2 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑅(𝐺𝑤)))
2417, 18, 19, 19, 20, 21, 22ofrfval 7409 . 2 (𝜑 → (𝐹r 𝑇𝐺 ↔ ∀𝑤𝐴 (𝐹𝑤)𝑇(𝐺𝑤)))
2516, 23, 243imtr4d 296 1 (𝜑 → (𝐹r 𝑅𝐺𝐹r 𝑇𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107  ∀wral 3136   class class class wbr 5057  ⟶wf 6344  ‘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 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  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 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  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: (None)
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