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Theorem caofrss 5677
Description: Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (φA 𝑉)
caofref.2 (φ𝐹:A𝑆)
caofcom.3 (φ𝐺:A𝑆)
caofrss.4 ((φ (x 𝑆 y 𝑆)) → (x𝑅yx𝑇y))
Assertion
Ref Expression
caofrss (φ → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
Distinct variable groups:   x,y,𝐹   x,𝐺,y   φ,x,y   x,𝑅,y   x,𝑆,y   x,𝑇,y
Allowed substitution hints:   A(x,y)   𝑉(x,y)

Proof of Theorem caofrss
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5 (φ𝐹:A𝑆)
21ffvelrnda 5245 . . . 4 ((φ w A) → (𝐹w) 𝑆)
3 caofcom.3 . . . . 5 (φ𝐺:A𝑆)
43ffvelrnda 5245 . . . 4 ((φ w A) → (𝐺w) 𝑆)
5 caofrss.4 . . . . . 6 ((φ (x 𝑆 y 𝑆)) → (x𝑅yx𝑇y))
65ralrimivva 2395 . . . . 5 (φx 𝑆 y 𝑆 (x𝑅yx𝑇y))
76adantr 261 . . . 4 ((φ w A) → x 𝑆 y 𝑆 (x𝑅yx𝑇y))
8 breq1 3758 . . . . . 6 (x = (𝐹w) → (x𝑅y ↔ (𝐹w)𝑅y))
9 breq1 3758 . . . . . 6 (x = (𝐹w) → (x𝑇y ↔ (𝐹w)𝑇y))
108, 9imbi12d 223 . . . . 5 (x = (𝐹w) → ((x𝑅yx𝑇y) ↔ ((𝐹w)𝑅y → (𝐹w)𝑇y)))
11 breq2 3759 . . . . . 6 (y = (𝐺w) → ((𝐹w)𝑅y ↔ (𝐹w)𝑅(𝐺w)))
12 breq2 3759 . . . . . 6 (y = (𝐺w) → ((𝐹w)𝑇y ↔ (𝐹w)𝑇(𝐺w)))
1311, 12imbi12d 223 . . . . 5 (y = (𝐺w) → (((𝐹w)𝑅y → (𝐹w)𝑇y) ↔ ((𝐹w)𝑅(𝐺w) → (𝐹w)𝑇(𝐺w))))
1410, 13rspc2va 2657 . . . 4 ((((𝐹w) 𝑆 (𝐺w) 𝑆) x 𝑆 y 𝑆 (x𝑅yx𝑇y)) → ((𝐹w)𝑅(𝐺w) → (𝐹w)𝑇(𝐺w)))
152, 4, 7, 14syl21anc 1133 . . 3 ((φ w A) → ((𝐹w)𝑅(𝐺w) → (𝐹w)𝑇(𝐺w)))
1615ralimdva 2381 . 2 (φ → (w A (𝐹w)𝑅(𝐺w) → w A (𝐹w)𝑇(𝐺w)))
17 ffn 4989 . . . 4 (𝐹:A𝑆𝐹 Fn A)
181, 17syl 14 . . 3 (φ𝐹 Fn A)
19 ffn 4989 . . . 4 (𝐺:A𝑆𝐺 Fn A)
203, 19syl 14 . . 3 (φ𝐺 Fn A)
21 caofref.1 . . 3 (φA 𝑉)
22 inidm 3140 . . 3 (AA) = A
23 eqidd 2038 . . 3 ((φ w A) → (𝐹w) = (𝐹w))
24 eqidd 2038 . . 3 ((φ w A) → (𝐺w) = (𝐺w))
2518, 20, 21, 21, 22, 23, 24ofrfval 5662 . 2 (φ → (𝐹𝑟 𝑅𝐺w A (𝐹w)𝑅(𝐺w)))
2618, 20, 21, 21, 22, 23, 24ofrfval 5662 . 2 (φ → (𝐹𝑟 𝑇𝐺w A (𝐹w)𝑇(𝐺w)))
2716, 25, 263imtr4d 192 1 (φ → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wral 2300   class class class wbr 3755   Fn wfn 4840  wf 4841  cfv 4845  𝑟 cofr 5653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ofr 5655
This theorem is referenced by: (None)
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