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Theorem bj-imafv 37252
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 2741 . . . 4 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ((𝐹 “ {𝐴}) = {𝑥} ↔ (𝐺 “ {𝐴}) = {𝑥}))
21abbidv 2808 . . 3 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
32unieqd 4920 . 2 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
4 dffv4 6903 . 2 (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
5 dffv4 6903 . 2 (𝐺𝐴) = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}}
63, 4, 53eqtr4g 2802 1 ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  {cab 2714  {csn 4626   cuni 4907  cima 5688  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569
This theorem is referenced by:  bj-funun  37253
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