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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 4716 | . . . 4 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝐷 = 𝐴) |
5 | 1, 4 | bj-fununsn1 34557 | . 2 ⊢ (𝜑 → (𝐺‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷)) |
6 | 2 | fvresd 6683 | . 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷)) |
7 | 5, 6 | eqtrd 2855 | 1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ∖ cdif 3926 ∪ cun 3927 {csn 4560 〈cop 4566 ↾ cres 5550 ‘cfv 6348 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 |
This theorem is referenced by: (None) |
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