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Mirrors > Home > MPE Home > Th. List > fnsnfvOLD | Structured version Visualization version GIF version |
Description: Obsolete version of fnsnfv 6768 as of 8-Aug-2024. (Contributed by NM, 22-May-1998.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
fnsnfvOLD | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2743 | . . . 4 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦) | |
2 | fnbrfvb 6743 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦)) | |
3 | 1, 2 | syl5bb 286 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑦 = (𝐹‘𝐵) ↔ 𝐵𝐹𝑦)) |
4 | 3 | abbidv 2800 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝑦 = (𝐹‘𝐵)} = {𝑦 ∣ 𝐵𝐹𝑦}) |
5 | df-sn 4528 | . . 3 ⊢ {(𝐹‘𝐵)} = {𝑦 ∣ 𝑦 = (𝐹‘𝐵)} | |
6 | 5 | a1i 11 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = {𝑦 ∣ 𝑦 = (𝐹‘𝐵)}) |
7 | fnrel 6458 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
8 | relimasn 5937 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) |
10 | 9 | adantr 484 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) |
11 | 4, 6, 10 | 3eqtr4d 2781 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cab 2714 {csn 4527 class class class wbr 5039 “ cima 5539 Rel wrel 5541 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: (None) |
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