Theorem bj-imafv 37818

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 2773	. . . 4 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ((𝐹 “ {𝐴}) = {𝑥} ↔ (𝐺 “ {𝐴}) = {𝑥}))
2	1	abbidv 2835	. . 3 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
3	2	unieqd 4887	. 2 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
4		dffv4 6879	. 2 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
5		dffv4 6879	. 2 ⊢ (𝐺‘𝐴) = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}}
6	3, 4, 5	3eqtr4g 2829	1 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:   → wi 4   = wceq 1567  {cab 2747  {csn 4592  ∪ cuni 4874   “ cima 5665  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fv 6545

This theorem is referenced by:  bj-funun  37819