Theorem bj-imafv 36861

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 2729	. . . 4 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ((𝐹 “ {𝐴}) = {𝑥} ↔ (𝐺 “ {𝐴}) = {𝑥}))
2	1	abbidv 2794	. . 3 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
3	2	unieqd 4922	. 2 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
4		dffv4 6893	. 2 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
5		dffv4 6893	. 2 ⊢ (𝐺‘𝐴) = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}}
6	3, 4, 5	3eqtr4g 2790	1 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:   → wi 4   = wceq 1533  {cab 2702  {csn 4630  ∪ cuni 4909   “ cima 5681  ‘cfv 6549

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 5300  ax-nul 5307  ax-pr 5429

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 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fv 6557

This theorem is referenced by:  bj-funun  36862