Theorem preimane 33043

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 18	. . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
5	4	necon3d 2981	. . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌}))
6	1, 5	mpd 16	. . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌})
7		preimane.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		funimacnv 6621	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
9	7, 8	syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
10	2	snssd 4754	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹)
11		dfss2 3924	. . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋})
12	10, 11	sylib 221	. . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋})
13	9, 12	eqtrd 2800	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋})
14		funimacnv 6621	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹))
15	7, 14	syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹))
16		preimane.1	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ran 𝐹)
17	16	snssd 4754	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹)
18		dfss2 3924	. . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌})
19	17, 18	sylib 221	. . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌})
20	15, 19	eqtrd 2800	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌})
21	6, 13, 20	3netr4d 3037	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌})))
22		imaeq2 6060	. . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌})))
23	22	necon3i 2992	. 2 ⊢ ((𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌})) → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌}))
24	21, 23	syl 18	1 ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∩ cin 3905   ⊆ wss 3906  {csn 4591  ^◡ccnv 5662  ran crn 5664   “ cima 5666  Fun wfun 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6542

This theorem is used by:  fnpreimac  33044