Theorem preimane 33311

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 4793	. . . . . 6 ⊢ (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
4	2, 3	syl 18	. . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
5	4	necon3d 2976	. . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌}))
6	1, 5	mpd 16	. . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌})
7		preimane.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		funimacnv 6622	. . . . 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 4743	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹)
11		dfss2 4094	. . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋})
12	10, 11	sylib 221	. . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋})
13	9, 12	eqtrd 2795	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋})
14		funimacnv 6622	. . . . 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 4743	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹)
18		dfss2 4094	. . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌})
19	17, 18	sylib 221	. . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌})
20	15, 19	eqtrd 2795	. . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌})
21	6, 13, 20	3netr4d 3032	. 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌})))
22		imaeq2 6047	. . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌})))
23	22	necon3i 2987	. 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 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∩ cin 4089  {csn 4584  ^◡ccnv 5649  ran crn 5651   “ cima 5653  Fun wfun 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6541

This theorem is used by:  fnpreimac  33312