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Theorem caofcan 37544
 Description: Transfer a cancellation law like mulcan 10543 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
Hypotheses
Ref Expression
caofcan.1 (𝜑𝐴𝑉)
caofcan.2 (𝜑𝐹:𝐴𝑇)
caofcan.3 (𝜑𝐺:𝐴𝑆)
caofcan.4 (𝜑𝐻:𝐴𝑆)
caofcan.5 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
Assertion
Ref Expression
caofcan (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem caofcan
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofcan.2 . . . . . . 7 (𝜑𝐹:𝐴𝑇)
2 ffn 5958 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
31, 2syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐴)
4 caofcan.3 . . . . . . 7 (𝜑𝐺:𝐴𝑆)
5 ffn 5958 . . . . . . 7 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
64, 5syl 17 . . . . . 6 (𝜑𝐺 Fn 𝐴)
7 caofcan.1 . . . . . 6 (𝜑𝐴𝑉)
8 inidm 3784 . . . . . 6 (𝐴𝐴) = 𝐴
9 eqidd 2611 . . . . . 6 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
10 eqidd 2611 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
113, 6, 7, 7, 8, 9, 10ofval 6804 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑤)𝑅(𝐺𝑤)))
12 caofcan.4 . . . . . . 7 (𝜑𝐻:𝐴𝑆)
13 ffn 5958 . . . . . . 7 (𝐻:𝐴𝑆𝐻 Fn 𝐴)
1412, 13syl 17 . . . . . 6 (𝜑𝐻 Fn 𝐴)
15 eqidd 2611 . . . . . 6 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
163, 14, 7, 7, 8, 9, 15ofval 6804 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐻)‘𝑤) = ((𝐹𝑤)𝑅(𝐻𝑤)))
1711, 16eqeq12d 2625 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤))))
18 simpl 472 . . . . 5 ((𝜑𝑤𝐴) → 𝜑)
191ffvelrnda 6267 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑇)
204ffvelrnda 6267 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
2112ffvelrnda 6267 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
22 caofcan.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
2322caovcang 6733 . . . . 5 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑇 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆)) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2418, 19, 20, 21, 23syl13anc 1320 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2517, 24bitrd 267 . . 3 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ (𝐺𝑤) = (𝐻𝑤)))
2625ralbidva 2968 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
273, 6, 7, 7, 8offn 6806 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝐴)
283, 14, 7, 7, 8offn 6806 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐻) Fn 𝐴)
29 eqfnfv 6219 . . 3 (((𝐹𝑓 𝑅𝐺) Fn 𝐴 ∧ (𝐹𝑓 𝑅𝐻) Fn 𝐴) → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
3027, 28, 29syl2anc 691 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
31 eqfnfv 6219 . . 3 ((𝐺 Fn 𝐴𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
326, 14, 31syl2anc 691 . 2 (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
3326, 30, 323bitr4d 299 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795 This theorem is referenced by: (None)
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