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Theorem funfvdm2f 5181
Description: The value of a function. Version of funfvdm2 5180 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
Hypotheses
Ref Expression
funfvdm2f.1 yA
funfvdm2f.2 y𝐹
Assertion
Ref Expression
funfvdm2f ((Fun 𝐹 A dom 𝐹) → (𝐹A) = {yA𝐹y})

Proof of Theorem funfvdm2f
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 funfvdm2 5180 . 2 ((Fun 𝐹 A dom 𝐹) → (𝐹A) = {wA𝐹w})
2 funfvdm2f.1 . . . . 5 yA
3 funfvdm2f.2 . . . . 5 y𝐹
4 nfcv 2175 . . . . 5 yw
52, 3, 4nfbr 3799 . . . 4 y A𝐹w
6 nfv 1418 . . . 4 w A𝐹y
7 breq2 3759 . . . 4 (w = y → (A𝐹wA𝐹y))
85, 6, 7cbvab 2157 . . 3 {wA𝐹w} = {yA𝐹y}
98unieqi 3581 . 2 {wA𝐹w} = {yA𝐹y}
101, 9syl6eq 2085 1 ((Fun 𝐹 A dom 𝐹) → (𝐹A) = {yA𝐹y})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162   cuni 3571   class class class wbr 3755  dom cdm 4288  Fun wfun 4839  cfv 4845
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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  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-v 2553  df-sbc 2759  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-br 3756  df-opab 3810  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-fv 4853
This theorem is referenced by: (None)
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