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Theorem caofcan 43631
Description: Transfer a cancellation law like mulcan 11850 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 6709 . . . . . 6 (𝜑𝐹 Fn 𝐴)
3 caofcan.3 . . . . . . 7 (𝜑𝐺:𝐴𝑆)
43ffnd 6709 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5 caofcan.1 . . . . . 6 (𝜑𝐴𝑉)
6 inidm 4211 . . . . . 6 (𝐴𝐴) = 𝐴
7 eqidd 2725 . . . . . 6 ((𝜑𝑤𝐴) → (𝐹𝑤) = (𝐹𝑤))
8 eqidd 2725 . . . . . 6 ((𝜑𝑤𝐴) → (𝐺𝑤) = (𝐺𝑤))
92, 4, 5, 5, 6, 7, 8ofval 7675 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹𝑤)𝑅(𝐺𝑤)))
10 caofcan.4 . . . . . . 7 (𝜑𝐻:𝐴𝑆)
1110ffnd 6709 . . . . . 6 (𝜑𝐻 Fn 𝐴)
12 eqidd 2725 . . . . . 6 ((𝜑𝑤𝐴) → (𝐻𝑤) = (𝐻𝑤))
132, 11, 5, 5, 6, 7, 12ofval 7675 . . . . 5 ((𝜑𝑤𝐴) → ((𝐹f 𝑅𝐻)‘𝑤) = ((𝐹𝑤)𝑅(𝐻𝑤)))
149, 13eqeq12d 2740 . . . 4 ((𝜑𝑤𝐴) → (((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤) ↔ ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐹𝑤)𝑅(𝐻𝑤))))
15 simpl 482 . . . . 5 ((𝜑𝑤𝐴) → 𝜑)
161ffvelcdmda 7077 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑇)
173ffvelcdmda 7077 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
1810ffvelcdmda 7077 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
19 caofcan.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))
2019caovcang 7602 . . . . 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 279 . . 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 7677 . . 3 (𝜑 → (𝐹f 𝑅𝐺) Fn 𝐴)
252, 11, 5, 5, 6offn 7677 . . 3 (𝜑 → (𝐹f 𝑅𝐻) Fn 𝐴)
26 eqfnfv 7023 . . 3 (((𝐹f 𝑅𝐺) Fn 𝐴 ∧ (𝐹f 𝑅𝐻) Fn 𝐴) → ((𝐹f 𝑅𝐺) = (𝐹f 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤)))
2724, 25, 26syl2anc 583 . 2 (𝜑 → ((𝐹f 𝑅𝐺) = (𝐹f 𝑅𝐻) ↔ ∀𝑤𝐴 ((𝐹f 𝑅𝐺)‘𝑤) = ((𝐹f 𝑅𝐻)‘𝑤)))
28 eqfnfv 7023 . . 3 ((𝐺 Fn 𝐴𝐻 Fn 𝐴) → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
294, 11, 28syl2anc 583 . 2 (𝜑 → (𝐺 = 𝐻 ↔ ∀𝑤𝐴 (𝐺𝑤) = (𝐻𝑤)))
3023, 27, 293bitr4d 311 1 (𝜑 → ((𝐹f 𝑅𝐺) = (𝐹f 𝑅𝐻) ↔ 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053   Fn wfn 6529  wf 6530  cfv 6534  (class class class)co 7402  f cof 7662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664
This theorem is referenced by: (None)
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