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Theorem dfimafn2 5437
Description: Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
dfimafn2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfimafn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfimafn 5436 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
2 iunab 3827 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
31, 2syl6eqr 2166 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦})
4 df-sn 3501 . . . . 5 {(𝐹𝑥)} = {𝑦𝑦 = (𝐹𝑥)}
5 eqcom 2117 . . . . . 6 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
65abbii 2231 . . . . 5 {𝑦𝑦 = (𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
74, 6eqtri 2136 . . . 4 {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
87a1i 9 . . 3 (𝑥𝐴 → {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦})
98iuneq2i 3799 . 2 𝑥𝐴 {(𝐹𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦}
103, 9syl6eqr 2166 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  {cab 2101  wrex 2392  wss 3039  {csn 3495   ciun 3781  dom cdm 4507  cima 4510  Fun wfun 5085  cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by: (None)
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