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Theorem caofcan 39492
Description: Transfer a cancellation law like mulcan 11015 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)
21ffnd 6294 . . . . . 6 (𝜑𝐹 Fn 𝐴)
3 caofcan.3 . . . . . . 7 (𝜑𝐺:𝐴𝑆)
43ffnd 6294 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5 caofcan.1 . . . . . 6 (𝜑𝐴𝑉)
6 inidm 4043 . . . . . 6 (𝐴𝐴) = 𝐴
7 eqidd 2779 . . . . . 6 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
8 eqidd 2779 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
92, 4, 5, 5, 6, 7, 8ofval 7185 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑤)𝑅(𝐺𝑤)))
10 caofcan.4 . . . . . . 7 (𝜑𝐻:𝐴𝑆)
1110ffnd 6294 . . . . . 6 (𝜑𝐻 Fn 𝐴)
12 eqidd 2779 . . . . . 6 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
132, 11, 5, 5, 6, 7, 12ofval 7185 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐻)‘𝑤) = ((𝐹𝑤)𝑅(𝐻𝑤)))
149, 13eqeq12d 2793 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤))))
15 simpl 476 . . . . 5 ((𝜑𝑤𝐴) → 𝜑)
161ffvelrnda 6625 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑇)
173ffvelrnda 6625 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
1810ffvelrnda 6625 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
19 caofcan.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
2019caovcang 7114 . . . . 5 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑇 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆)) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2115, 16, 17, 18, 20syl13anc 1440 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2214, 21bitrd 271 . . 3 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ (𝐺𝑤) = (𝐻𝑤)))
2322ralbidva 3167 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
242, 4, 5, 5, 6offn 7187 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝐴)
252, 11, 5, 5, 6offn 7187 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐻) Fn 𝐴)
26 eqfnfv 6576 . . 3 (((𝐹𝑓 𝑅𝐺) Fn 𝐴 ∧ (𝐹𝑓 𝑅𝐻) Fn 𝐴) → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
2724, 25, 26syl2anc 579 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
28 eqfnfv 6576 . . 3 ((𝐺 Fn 𝐴𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
294, 11, 28syl2anc 579 . 2 (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
3023, 27, 293bitr4d 303 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090   Fn wfn 6132  wf 6133  cfv 6137  (class class class)co 6924  𝑓 cof 7174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176
This theorem is referenced by: (None)
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