Theorem preimane 32648

Description: Different elements have different preimages. (Contributed by Thierry Arnoux, 7-May-2023.)

Hypotheses

Ref	Expression
preimane.f	⊢ (𝜑 → Fun 𝐹)
preimane.x	⊢ (𝜑 → 𝑋 ≠ 𝑌)
preimane.y	⊢ (𝜑 → 𝑋 ∈ ran 𝐹)
preimane.1	⊢ (𝜑 → 𝑌 ∈ ran 𝐹)

Assertion

Ref	Expression
preimane	⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌}))

Proof of Theorem preimane

Step	Hyp	Ref	Expression
1		preimane.x	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
2		preimane.y	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ran 𝐹)
3		sneqrg 4815	. . . . . 6 ⊢ (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
5	4	necon3d 2953	. . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌}))
6	1, 5	mpd 15	. . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌})
7		preimane.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		funimacnv 6617	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
10	2	snssd 4785	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹)
11		dfss2 3944	. . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋})
12	10, 11	sylib 218	. . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋})
13	9, 12	eqtrd 2770	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋})
14		funimacnv 6617	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹))
15	7, 14	syl 17	. . . 4 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹))
16		preimane.1	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ran 𝐹)
17	16	snssd 4785	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹)
18		dfss2 3944	. . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌})
19	17, 18	sylib 218	. . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌})
20	15, 19	eqtrd 2770	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌})
21	6, 13, 20	3netr4d 3009	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌})))
22		imaeq2 6043	. . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌})))
23	22	necon3i 2964	. 2 ⊢ ((𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌})) → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌}))
24	21, 23	syl 17	1 ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   ∩ cin 3925   ⊆ wss 3926  {csn 4601  ^◡ccnv 5653  ran crn 5655   “ cima 5657  Fun wfun 6525

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-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6533

This theorem is referenced by:  fnpreimac  32649