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Theorem bj-imafv 36787
Description: If the direct image of a singleton under any of two functions is the same, then the values of these functions at the corresponding point agree. (Contributed by BJ, 18-Mar-2023.)
Assertion
Ref Expression
bj-imafv ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem bj-imafv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2729 . . . 4 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ((𝐹 “ {𝐴}) = {𝑥} ↔ (𝐺 “ {𝐴}) = {𝑥}))
21abbidv 2794 . . 3 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
32unieqd 4916 . 2 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
4 dffv4 6889 . 2 (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
5 dffv4 6889 . 2 (𝐺𝐴) = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}}
63, 4, 53eqtr4g 2790 1 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  {cab 2702  {csn 4624   cuni 4903  cima 5675  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fv 6551
This theorem is referenced by:  bj-funun  36788
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