Theorem preimane 32922

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 4799	. . . . . 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 6606	. . . . 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 4748	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹)
11		dfss2 3925	. . . . 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 6606	. . . . 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 4748	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹)
18		dfss2 3925	. . . . 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 6048	. . 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}))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   ∩ cin 3906   ⊆ wss 3907  {csn 4585  ^◡ccnv 5650  ran crn 5652   “ cima 5654  Fun wfun 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6527

This theorem is referenced by:  fnpreimac  32923