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Theorem caofcan 40948
Description: Transfer a cancellation law like mulcan 11276 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 6505 . . . . . 6 (𝜑𝐹 Fn 𝐴)
3 caofcan.3 . . . . . . 7 (𝜑𝐺:𝐴𝑆)
43ffnd 6505 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5 caofcan.1 . . . . . 6 (𝜑𝐴𝑉)
6 inidm 4181 . . . . . 6 (𝐴𝐴) = 𝐴
7 eqidd 2825 . . . . . 6 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
8 eqidd 2825 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
92, 4, 5, 5, 6, 7, 8ofval 7413 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹𝑤)𝑅(𝐺𝑤)))
10 caofcan.4 . . . . . . 7 (𝜑𝐻:𝐴𝑆)
1110ffnd 6505 . . . . . 6 (𝜑𝐻 Fn 𝐴)
12 eqidd 2825 . . . . . 6 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
132, 11, 5, 5, 6, 7, 12ofval 7413 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹f 𝑅𝐻)‘𝑤) = ((𝐹𝑤)𝑅(𝐻𝑤)))
149, 13eqeq12d 2840 . . . 4 ((𝜑𝑤𝐴) → (((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤) ↔ ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤))))
15 simpl 486 . . . . 5 ((𝜑𝑤𝐴) → 𝜑)
161ffvelrnda 6843 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑇)
173ffvelrnda 6843 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
1810ffvelrnda 6843 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
19 caofcan.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
2019caovcang 7344 . . . . 5 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑇 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆)) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2115, 16, 17, 18, 20syl13anc 1369 . . . 4 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤)) ↔ (𝐺𝑤) = (𝐻𝑤)))
2214, 21bitrd 282 . . 3 ((𝜑𝑤𝐴) → (((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤) ↔ (𝐺𝑤) = (𝐻𝑤)))
2322ralbidva 3191 . 2 (𝜑 → (∀𝑤𝐴 ((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤) ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
242, 4, 5, 5, 6offn 7415 . . 3 (𝜑 → (𝐹f 𝑅𝐺) Fn 𝐴)
252, 11, 5, 5, 6offn 7415 . . 3 (𝜑 → (𝐹f 𝑅𝐻) Fn 𝐴)
26 eqfnfv 6794 . . 3 (((𝐹f 𝑅𝐺) Fn 𝐴 ∧ (𝐹f 𝑅𝐻) Fn 𝐴) → ((𝐹f 𝑅𝐺) = (𝐹f 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤)))
2724, 25, 26syl2anc 587 . 2 (𝜑 → ((𝐹f 𝑅𝐺) = (𝐹f 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤)))
28 eqfnfv 6794 . . 3 ((𝐺 Fn 𝐴𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
294, 11, 28syl2anc 587 . 2 (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
3023, 27, 293bitr4d 314 1 (𝜑 → ((𝐹f 𝑅𝐺) = (𝐹f 𝑅𝐻) ↔ 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133   Fn wfn 6339  wf 6340  cfv 6344  (class class class)co 7150  f cof 7402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7404
This theorem is referenced by: (None)
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