Theorem bj-imafv 37425

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}) → (𝐹‘𝐴) = (𝐺‘𝐴))

Proof of Theorem bj-imafv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2739	. . . 4 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ((𝐹 “ {𝐴}) = {𝑥} ↔ (𝐺 “ {𝐴}) = {𝑥}))
2	1	abbidv 2801	. . 3 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
3	2	unieqd 4875	. 2 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
4		dffv4 6830	. 2 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
5		dffv4 6830	. 2 ⊢ (𝐺‘𝐴) = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}}
6	3, 4, 5	3eqtr4g 2795	1 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:   → wi 4   = wceq 1542  {cab 2713  {csn 4579  ∪ cuni 4862   “ cima 5626  ‘cfv 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fv 6499

This theorem is referenced by:  bj-funun  37426