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Theorem ofrfval2 6957
 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 2995 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
3 eqid 2651 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6058 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 offval2.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6019 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 247 . . 3 (𝜑𝐹 Fn 𝐴)
9 offval2.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝑋)
109ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
11 eqid 2651 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1211fnmpt 6058 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1310, 12syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
14 offval2.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1514fneq1d 6019 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1613, 15mpbird 247 . . 3 (𝜑𝐺 Fn 𝐴)
17 offval2.1 . . 3 (𝜑𝐴𝑉)
18 inidm 3855 . . 3 (𝐴𝐴) = 𝐴
196adantr 480 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2019fveq1d 6231 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2114adantr 480 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2221fveq1d 6231 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
238, 16, 17, 17, 18, 20, 22ofrfval 6947 . 2 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)))
24 nffvmpt1 6237 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
25 nfcv 2793 . . . . 5 𝑥𝑅
26 nffvmpt1 6237 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
2724, 25, 26nfbr 4732 . . . 4 𝑥((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)
28 nfv 1883 . . . 4 𝑦((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)
29 fveq2 6229 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
30 fveq2 6229 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3129, 30breq12d 4698 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3227, 28, 31cbvral 3197 . . 3 (∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
33 simpr 476 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
343fvmpt2 6330 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3533, 1, 34syl2anc 694 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3611fvmpt2 6330 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3733, 9, 36syl2anc 694 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3835, 37breq12d 4698 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥) ↔ 𝐵𝑅𝐶))
3938ralbidva 3014 . . 3 (𝜑 → (∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥) ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
4032, 39syl5bb 272 . 2 (𝜑 → (∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
4123, 40bitrd 268 1 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   class class class wbr 4685   ↦ cmpt 4762   Fn wfn 5921  ‘cfv 5926   ∘𝑟 cofr 6938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ofr 6940 This theorem is referenced by:  gsumbagdiaglem  19423  mplmonmul  19512  coe1mul2lem1  19685  itg2const  23552  itg2const2  23553  itg2uba  23555  itg2mulclem  23558  itg2splitlem  23560  itg2split  23561  itg2monolem1  23562  itg2gt0  23572  itg2cnlem1  23573  itg2cnlem2  23574  iblss  23616  i1fibl  23619  itgitg1  23620  itgle  23621  ibladdlem  23631  iblabs  23640  iblabsr  23641  iblmulc2  23642  bddmulibl  23650  itg2addnclem  33591  itg2addnclem3  33593  itg2addnc  33594  itg2gt0cn  33595  ibladdnclem  33596  iblabsnc  33604  iblmulc2nc  33605  bddiblnc  33610  ftc1anclem4  33618  ftc1anclem5  33619  ftc1anclem6  33620  ftc1anclem7  33621  ftc1anclem8  33622  ftc1anc  33623
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