Theorem fvf1pr 7327

Description: Values of a one-to-one function between two sets with two elements. Actually, such a function is a bijection. (Contributed by AV, 22-Jul-2025.)

Assertion

Ref	Expression
fvf1pr	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))

Proof of Theorem fvf1pr

Step	Hyp	Ref	Expression
1		f1f 6805	. . 3 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → 𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌})
2		prid1g 4765	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
3	2	3ad2ant1 1132	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ {𝐴, 𝐵})
4		ffvelcdm 7101	. . 3 ⊢ ((𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌} ∧ 𝐴 ∈ {𝐴, 𝐵}) → (𝐹‘𝐴) ∈ {𝑋, 𝑌})
5	1, 3, 4	syl2anr 597	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (𝐹‘𝐴) ∈ {𝑋, 𝑌})
6		prid2g 4766	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
7	6	3ad2ant2 1133	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ {𝐴, 𝐵})
8		ffvelcdm 7101	. . 3 ⊢ ((𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌} ∧ 𝐵 ∈ {𝐴, 𝐵}) → (𝐹‘𝐵) ∈ {𝑋, 𝑌})
9	1, 7, 8	syl2anr 597	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (𝐹‘𝐵) ∈ {𝑋, 𝑌})
10		elpri 4654	. . 3 ⊢ ((𝐹‘𝐴) ∈ {𝑋, 𝑌} → ((𝐹‘𝐴) = 𝑋 ∨ (𝐹‘𝐴) = 𝑌))
11		elpri 4654	. . 3 ⊢ ((𝐹‘𝐵) ∈ {𝑋, 𝑌} → ((𝐹‘𝐵) = 𝑋 ∨ (𝐹‘𝐵) = 𝑌))
12		eqtr3 2761	. . . . . . . 8 ⊢ (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
13	3, 7	jca 511	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵}))
14		f1veqaeq 7277	. . . . . . . . 9 ⊢ ((𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
15	13, 14	sylan2 593	. . . . . . . 8 ⊢ ((𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
16	12, 15	syl5 34	. . . . . . 7 ⊢ ((𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → 𝐴 = 𝐵))
17	16	ex 412	. . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → 𝐴 = 𝐵)))
18		eqneqall 2949	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
19	18	com12 32	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
20	19	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
21	20	a1i 11	. . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))))
22	17, 21	syldd 72	. . . . 5 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))))
23	22	impcom 407	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
24		olc 868	. . . . 5 ⊢ (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))
25	24	a1i 11	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
26		orc 867	. . . . 5 ⊢ (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))
27	26	a1i 11	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
28		eqtr3 2761	. . . . . . . 8 ⊢ (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑌) → (𝐹‘𝐴) = (𝐹‘𝐵))
29	28, 15	syl5 34	. . . . . . 7 ⊢ ((𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑌) → 𝐴 = 𝐵))
30	29	ex 412	. . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑌) → 𝐴 = 𝐵)))
31	30, 21	syldd 72	. . . . 5 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑌) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))))
32	31	impcom 407	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑌) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
33	23, 25, 27, 32	ccased 1038	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → ((((𝐹‘𝐴) = 𝑋 ∨ (𝐹‘𝐴) = 𝑌) ∧ ((𝐹‘𝐵) = 𝑋 ∨ (𝐹‘𝐵) = 𝑌)) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
34	10, 11, 33	syl2ani 607	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) ∈ {𝑋, 𝑌} ∧ (𝐹‘𝐵) ∈ {𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
35	5, 9, 34	mp2and 699	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  {cpr 4633  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571

This theorem is referenced by: (None)