Theorem fvf1pr 7315

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 6778	. . 3 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → 𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌})
2		prid1g 4720	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
3	2	3ad2ant1 1151	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ {𝐴, 𝐵})
4		ffvelcdm 7081	. . 3 ⊢ ((𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌} ∧ 𝐴 ∈ {𝐴, 𝐵}) → (𝐹‘𝐴) ∈ {𝑋, 𝑌})
5	1, 3, 4	syl2anr 609	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (𝐹‘𝐴) ∈ {𝑋, 𝑌})
6		prid2g 4721	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
7	6	3ad2ant2 1152	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ {𝐴, 𝐵})
8		ffvelcdm 7081	. . 3 ⊢ ((𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌} ∧ 𝐵 ∈ {𝐴, 𝐵}) → (𝐹‘𝐵) ∈ {𝑋, 𝑌})
9	1, 7, 8	syl2anr 609	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (𝐹‘𝐵) ∈ {𝑋, 𝑌})
10		elpri 4607	. . 3 ⊢ ((𝐹‘𝐴) ∈ {𝑋, 𝑌} → ((𝐹‘𝐴) = 𝑋 ∨ (𝐹‘𝐴) = 𝑌))
11		elpri 4607	. . 3 ⊢ ((𝐹‘𝐵) ∈ {𝑋, 𝑌} → ((𝐹‘𝐵) = 𝑋 ∨ (𝐹‘𝐵) = 𝑌))
12		eqtr3 2782	. . . . . . . 8 ⊢ (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
13	3, 7	jca 521	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵}))
14		f1veqaeq 7260	. . . . . . . . 9 ⊢ ((𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
15	13, 14	sylan2 605	. . . . . . . 8 ⊢ ((𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
16	12, 15	syl5 35	. . . . . . 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 418	. . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → 𝐴 = 𝐵)))
18		eqneqall 2966	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
19	18	com12 33	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋))))
20	19	3ad2ant3 1153	. . . . . . 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 73	. . . . 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 413	. . . 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 882	. . . . 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 881	. . . . 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 2782	. . . . . . . 8 ⊢ (((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑌) → (𝐹‘𝐴) = (𝐹‘𝐵))
29	28, 15	syl5 35	. . . . . . 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 418	. . . . . 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 73	. . . . 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 413	. . . 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 1054	. . 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 619	. 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 712	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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  {cpr 4585  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538

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-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  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-nf 1816  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 4584  df-pr 4586  df-op 4590  df-uni 4867  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-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546

This theorem is used by: (None)