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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsetpreimafveqfv | Structured version Visualization version GIF version |
Description: The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.) |
Ref | Expression |
---|---|
setpreimafvex.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
Ref | Expression |
---|---|
elsetpreimafveqfv | ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝐹‘𝑋) = (𝐹‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setpreimafvex.p | . . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
2 | 1 | elsetpreimafvbi 43626 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))) |
3 | simpr 487 | . . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)) → (𝐹‘𝑌) = (𝐹‘𝑋)) | |
4 | 3 | eqcomd 2826 | . . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)) → (𝐹‘𝑋) = (𝐹‘𝑌)) |
5 | 2, 4 | syl6bi 255 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 → (𝐹‘𝑋) = (𝐹‘𝑌))) |
6 | 5 | 3exp 1114 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝑆 ∈ 𝑃 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 → (𝐹‘𝑋) = (𝐹‘𝑌))))) |
7 | 6 | 3imp2 1344 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝐹‘𝑋) = (𝐹‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 {cab 2798 ∃wrex 3138 {csn 4560 ◡ccnv 5547 “ cima 5551 Fn wfn 6343 ‘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-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-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: imasetpreimafvbijlemfv 43637 |
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