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Theorem caofcan 37441
Description: Transfer a cancellation law like mulcan 10413 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 5843 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
31, 2syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐴)
4 caofcan.3 . . . . . . 7 (𝜑𝐺:𝐴𝑆)
5 ffn 5843 . . . . . . 7 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
64, 5syl 17 . . . . . 6 (𝜑𝐺 Fn 𝐴)
7 caofcan.1 . . . . . 6 (𝜑𝐴𝑉)
8 inidm 3687 . . . . . 6 (𝐴𝐴) = 𝐴
9 eqidd 2515 . . . . . 6 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
10 eqidd 2515 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
113, 6, 7, 7, 8, 9, 10ofval 6680 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑤)𝑅(𝐺𝑤)))
12 caofcan.4 . . . . . . 7 (𝜑𝐻:𝐴𝑆)
13 ffn 5843 . . . . . . 7 (𝐻:𝐴𝑆𝐻 Fn 𝐴)
1412, 13syl 17 . . . . . 6 (𝜑𝐻 Fn 𝐴)
15 eqidd 2515 . . . . . 6 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
163, 14, 7, 7, 8, 9, 15ofval 6680 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐻)‘𝑤) = ((𝐹𝑤)𝑅(𝐻𝑤)))
1711, 16eqeq12d 2529 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤))))
18 simpl 471 . . . . 5 ((𝜑𝑤𝐴) → 𝜑)
191ffvelrnda 6151 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑇)
204ffvelrnda 6151 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
2112ffvelrnda 6151 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
22 caofcan.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
2322caovcang 6609 . . . . 5 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑇 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆)) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2418, 19, 20, 21, 23syl13anc 1319 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2517, 24bitrd 266 . . 3 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ (𝐺𝑤) = (𝐻𝑤)))
2625ralbidva 2872 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
273, 6, 7, 7, 8offn 6682 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝐴)
283, 14, 7, 7, 8offn 6682 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐻) Fn 𝐴)
29 eqfnfv 6103 . . 3 (((𝐹𝑓 𝑅𝐺) Fn 𝐴 ∧ (𝐹𝑓 𝑅𝐻) Fn 𝐴) → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
3027, 28, 29syl2anc 690 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
31 eqfnfv 6103 . . 3 ((𝐺 Fn 𝐴𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
326, 14, 31syl2anc 690 . 2 (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
3326, 30, 323bitr4d 298 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800   Fn wfn 5684  wf 5685  cfv 5689  (class class class)co 6426  𝑓 cof 6669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-of 6671
This theorem is referenced by: (None)
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