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Theorem ofrfval2 5754
Description: The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval2.1 (𝜑𝐴𝑉)
offval2.2 ((𝜑𝑥𝐴) → 𝐵𝑊)
offval2.3 ((𝜑𝑥𝐴) → 𝐶𝑋)
offval2.4 (𝜑𝐹 = (𝑥𝐴𝐵))
offval2.5 (𝜑𝐺 = (𝑥𝐴𝐶))
Assertion
Ref Expression
ofrfval2 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem ofrfval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 offval2.2 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑊)
21ralrimiva 2409 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
3 eqid 2056 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 5052 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 14 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 offval2.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 5016 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 160 . . 3 (𝜑𝐹 Fn 𝐴)
9 offval2.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝑋)
109ralrimiva 2409 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
11 eqid 2056 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1211fnmpt 5052 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1310, 12syl 14 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
14 offval2.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1514fneq1d 5016 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1613, 15mpbird 160 . . 3 (𝜑𝐺 Fn 𝐴)
17 offval2.1 . . 3 (𝜑𝐴𝑉)
18 inidm 3173 . . 3 (𝐴𝐴) = 𝐴
196adantr 265 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2019fveq1d 5207 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2114adantr 265 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2221fveq1d 5207 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
238, 16, 17, 17, 18, 20, 22ofrfval 5747 . 2 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)))
24 nffvmpt1 5213 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
25 nfcv 2194 . . . . 5 𝑥𝑅
26 nffvmpt1 5213 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
2724, 25, 26nfbr 3835 . . . 4 𝑥((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)
28 nfv 1437 . . . 4 𝑦((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)
29 fveq2 5205 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
30 fveq2 5205 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3129, 30breq12d 3804 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3227, 28, 31cbvral 2546 . . 3 (∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
33 simpr 107 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
343fvmpt2 5281 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3533, 1, 34syl2anc 397 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3611fvmpt2 5281 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3733, 9, 36syl2anc 397 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3835, 37breq12d 3804 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥) ↔ 𝐵𝑅𝐶))
3938ralbidva 2339 . . 3 (𝜑 → (∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥) ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
4032, 39syl5bb 185 . 2 (𝜑 → (∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
4123, 40bitrd 181 1 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323   class class class wbr 3791  cmpt 3845   Fn wfn 4924  cfv 4929  𝑟 cofr 5738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ofr 5740
This theorem is referenced by: (None)
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