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Theorem caofcan 39042
Description: Transfer a cancellation law like mulcan 10876 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 6206 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
31, 2syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐴)
4 caofcan.3 . . . . . . 7 (𝜑𝐺:𝐴𝑆)
5 ffn 6206 . . . . . . 7 (𝐺:𝐴𝑆𝐺 Fn 𝐴)
64, 5syl 17 . . . . . 6 (𝜑𝐺 Fn 𝐴)
7 caofcan.1 . . . . . 6 (𝜑𝐴𝑉)
8 inidm 3965 . . . . . 6 (𝐴𝐴) = 𝐴
9 eqidd 2761 . . . . . 6 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
10 eqidd 2761 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
113, 6, 7, 7, 8, 9, 10ofval 7072 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑤)𝑅(𝐺𝑤)))
12 caofcan.4 . . . . . . 7 (𝜑𝐻:𝐴𝑆)
13 ffn 6206 . . . . . . 7 (𝐻:𝐴𝑆𝐻 Fn 𝐴)
1412, 13syl 17 . . . . . 6 (𝜑𝐻 Fn 𝐴)
15 eqidd 2761 . . . . . 6 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
163, 14, 7, 7, 8, 9, 15ofval 7072 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹𝑓 𝑅𝐻)‘𝑤) = ((𝐹𝑤)𝑅(𝐻𝑤)))
1711, 16eqeq12d 2775 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤))))
18 simpl 474 . . . . 5 ((𝜑𝑤𝐴) → 𝜑)
191ffvelrnda 6523 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑇)
204ffvelrnda 6523 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
2112ffvelrnda 6523 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
22 caofcan.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
2322caovcang 7001 . . . . 5 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑇 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆)) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2418, 19, 20, 21, 23syl13anc 1479 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2517, 24bitrd 268 . . 3 ((𝜑𝑤𝐴) → (((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ (𝐺𝑤) = (𝐻𝑤)))
2625ralbidva 3123 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤) ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
273, 6, 7, 7, 8offn 7074 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝐴)
283, 14, 7, 7, 8offn 7074 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐻) Fn 𝐴)
29 eqfnfv 6475 . . 3 (((𝐹𝑓 𝑅𝐺) Fn 𝐴 ∧ (𝐹𝑓 𝑅𝐻) Fn 𝐴) → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
3027, 28, 29syl2anc 696 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹𝑓 𝑅𝐺)‘𝑤) = ((𝐹𝑓 𝑅𝐻)‘𝑤)))
31 eqfnfv 6475 . . 3 ((𝐺 Fn 𝐴𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
326, 14, 31syl2anc 696 . 2 (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
3326, 30, 323bitr4d 300 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  𝑓 cof 7061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063
This theorem is referenced by: (None)
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