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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fvsnun1 | Structured version Visualization version GIF version |
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. (Contributed by NM, 23-Sep-2007.) Put in deduction form and remove two sethood hypotheses. (Revised by BJ, 18-Mar-2023.) |
Ref | Expression |
---|---|
bj-fvsnun.un | ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩})) |
bj-fvsnun1.eldif | ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴})) |
Ref | Expression |
---|---|
bj-fvsnun1 | ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-fvsnun.un | . . 3 ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩})) | |
2 | bj-fvsnun1.eldif | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴})) | |
3 | eldifsnneq 4790 | . . . 4 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝐷 = 𝐴) |
5 | 1, 4 | bj-fununsn1 36788 | . 2 ⊢ (𝜑 → (𝐺‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)) |
6 | 2 | fvresd 6911 | . 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷)) |
7 | 5, 6 | eqtrd 2765 | 1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3937 ∪ cun 3938 {csn 4624 ⟨cop 4630 ↾ cres 5674 ‘cfv 6542 |
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 5294 ax-nul 5301 ax-pr 5423 |
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-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fv 6550 |
This theorem is referenced by: (None) |
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