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Theorem ofrfval2 7638
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 3143 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
3 eqid 2736 . . . . . 6 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6641 . . . . 5 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 offval2.4 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6595 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 256 . . 3 (𝜑𝐹 Fn 𝐴)
9 offval2.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝑋)
109ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝑋)
11 eqid 2736 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
1211fnmpt 6641 . . . . 5 (∀𝑥𝐴 𝐶𝑋 → (𝑥𝐴𝐶) Fn 𝐴)
1310, 12syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
14 offval2.5 . . . . 5 (𝜑𝐺 = (𝑥𝐴𝐶))
1514fneq1d 6595 . . . 4 (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
1613, 15mpbird 256 . . 3 (𝜑𝐺 Fn 𝐴)
17 offval2.1 . . 3 (𝜑𝐴𝑉)
18 inidm 4178 . . 3 (𝐴𝐴) = 𝐴
196adantr 481 . . . 4 ((𝜑𝑦𝐴) → 𝐹 = (𝑥𝐴𝐵))
2019fveq1d 6844 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) = ((𝑥𝐴𝐵)‘𝑦))
2114adantr 481 . . . 4 ((𝜑𝑦𝐴) → 𝐺 = (𝑥𝐴𝐶))
2221fveq1d 6844 . . 3 ((𝜑𝑦𝐴) → (𝐺𝑦) = ((𝑥𝐴𝐶)‘𝑦))
238, 16, 17, 17, 18, 20, 22ofrfval 7627 . 2 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)))
24 nffvmpt1 6853 . . . . 5 𝑥((𝑥𝐴𝐵)‘𝑦)
25 nfcv 2907 . . . . 5 𝑥𝑅
26 nffvmpt1 6853 . . . . 5 𝑥((𝑥𝐴𝐶)‘𝑦)
2724, 25, 26nfbr 5152 . . . 4 𝑥((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦)
28 nfv 1917 . . . 4 𝑦((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)
29 fveq2 6842 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
30 fveq2 6842 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐶)‘𝑦) = ((𝑥𝐴𝐶)‘𝑥))
3129, 30breq12d 5118 . . . 4 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥)))
3227, 28, 31cbvralw 3289 . . 3 (∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥))
33 simpr 485 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
343fvmpt2 6959 . . . . . 6 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3533, 1, 34syl2anc 584 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3611fvmpt2 6959 . . . . . 6 ((𝑥𝐴𝐶𝑋) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3733, 9, 36syl2anc 584 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
3835, 37breq12d 5118 . . . 4 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥) ↔ 𝐵𝑅𝐶))
3938ralbidva 3172 . . 3 (𝜑 → (∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥)𝑅((𝑥𝐴𝐶)‘𝑥) ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
4032, 39bitrid 282 . 2 (𝜑 → (∀𝑦𝐴 ((𝑥𝐴𝐵)‘𝑦)𝑅((𝑥𝐴𝐶)‘𝑦) ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
4123, 40bitrd 278 1 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064   class class class wbr 5105  cmpt 5188   Fn wfn 6491  cfv 6496  r cofr 7616
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ofr 7618
This theorem is referenced by:  gsumbagdiaglemOLD  21340  gsumbagdiaglem  21343  mplmonmul  21437  coe1mul2lem1  21638  itg2const  25105  itg2const2  25106  itg2uba  25108  itg2mulclem  25111  itg2splitlem  25113  itg2split  25114  itg2monolem1  25115  itg2gt0  25125  itg2cnlem1  25126  itg2cnlem2  25127  iblss  25169  i1fibl  25172  itgitg1  25173  itgle  25174  ibladdlem  25184  iblabs  25193  iblabsr  25194  iblmulc2  25195  bddmulibl  25203  bddiblnc  25206  itg2addnclem  36129  itg2addnclem3  36131  itg2addnc  36132  itg2gt0cn  36133  ibladdnclem  36134  iblabsnc  36142  iblmulc2nc  36143  ftc1anclem4  36154  ftc1anclem5  36155  ftc1anclem6  36156  ftc1anclem7  36157  ftc1anclem8  36158  ftc1anc  36159
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