Theorem fvf1pr 7236

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 6715	. . 3 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → 𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌})
2		prid1g 4711	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
3	2	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ {𝐴, 𝐵})
4		ffvelcdm 7009	. . 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 4712	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
7	6	3ad2ant2 1134	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ {𝐴, 𝐵})
8		ffvelcdm 7009	. . 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 4598	. . 3 ⊢ ((𝐹‘𝐴) ∈ {𝑋, 𝑌} → ((𝐹‘𝐴) = 𝑋 ∨ (𝐹‘𝐴) = 𝑌))
11		elpri 4598	. . 3 ⊢ ((𝐹‘𝐵) ∈ {𝑋, 𝑌} → ((𝐹‘𝐵) = 𝑋 ∨ (𝐹‘𝐵) = 𝑌))
12		eqtr3 2752	. . . . . . . 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 7185	. . . . . . . . 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 2937	. . . . . . . . 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 1135	. . . . . . 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 2752	. . . . . . . 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 1541   ∈ wcel 2110   ≠ wne 2926  {cpr 4576  ⟶wf 6473  –1-1→wf1 6474  ‘cfv 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fv 6485

This theorem is referenced by: (None)