Theorem preimane 32601

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 4806	. . . . . 6 ⊢ (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
5	4	necon3d 2947	. . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌}))
6	1, 5	mpd 15	. . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌})
7		preimane.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		funimacnv 6600	. . . . 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 4776	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹)
11		dfss2 3935	. . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋})
12	10, 11	sylib 218	. . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋})
13	9, 12	eqtrd 2765	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋})
14		funimacnv 6600	. . . . 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 4776	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹)
18		dfss2 3935	. . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌})
19	17, 18	sylib 218	. . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌})
20	15, 19	eqtrd 2765	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌})
21	6, 13, 20	3netr4d 3003	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌})))
22		imaeq2 6030	. . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌})))
23	22	necon3i 2958	. 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 2109   ≠ wne 2926   ∩ cin 3916   ⊆ wss 3917  {csn 4592  ^◡ccnv 5640  ran crn 5642   “ cima 5644  Fun wfun 6508

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 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-fun 6516

This theorem is referenced by:  fnpreimac  32602