Theorem bj-imafv 38363

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 2767	. . . 4 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ((𝐹 “ {𝐴}) = {𝑥} ↔ (𝐺 “ {𝐴}) = {𝑥}))
2	1	abbidv 2829	. . 3 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
3	2	unieqd 4883	. 2 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}})
4		dffv4 6902	. 2 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
5		dffv4 6902	. 2 ⊢ (𝐺‘𝐴) = ∪ {𝑥 ∣ (𝐺 “ {𝐴}) = {𝑥}}
6	3, 4, 5	3eqtr4g 2823	1 ⊢ ((𝐹 “ {𝐴}) = (𝐺 “ {𝐴}) → (𝐹‘𝐴) = (𝐺‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {cab 2741  {csn 4588  ∪ cuni 4870   “ cima 5658  ‘cfv 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fv 6563

This theorem is used by:  bj-funun  38364