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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 4789 | . . . 4 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝐷 = 𝐴) |
5 | 1, 4 | bj-fununsn1 37232 | . 2 ⊢ (𝜑 → (𝐺‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)) |
6 | 2 | fvresd 6924 | . 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷)) |
7 | 5, 6 | eqtrd 2776 | 1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3947 ∪ cun 3948 {csn 4624 〈cop 4630 ↾ cres 5685 ‘cfv 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-xp 5689 df-cnv 5691 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fv 6567 |
This theorem is referenced by: (None) |
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