Theorem fvf1pr 7305

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 6774	. . 3 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → 𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌})
2		prid1g 4725	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
3	2	3ad2ant1 1149	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ {𝐴, 𝐵})
4		ffvelcdm 7076	. . 3 ⊢ ((𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌} ∧ 𝐴 ∈ {𝐴, 𝐵}) → (𝐹‘𝐴) ∈ {𝑋, 𝑌})
5	1, 3, 4	syl2anr 608	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (𝐹‘𝐴) ∈ {𝑋, 𝑌})
6		prid2g 4726	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
7	6	3ad2ant2 1150	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ {𝐴, 𝐵})
8		ffvelcdm 7076	. . 3 ⊢ ((𝐹:{𝐴, 𝐵}⟶{𝑋, 𝑌} ∧ 𝐵 ∈ {𝐴, 𝐵}) → (𝐹‘𝐵) ∈ {𝑋, 𝑌})
9	1, 7, 8	syl2anr 608	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (𝐹‘𝐵) ∈ {𝑋, 𝑌})
10		elpri 4612	. . 3 ⊢ ((𝐹‘𝐴) ∈ {𝑋, 𝑌} → ((𝐹‘𝐴) = 𝑋 ∨ (𝐹‘𝐴) = 𝑌))
11		elpri 4612	. . 3 ⊢ ((𝐹‘𝐵) ∈ {𝑋, 𝑌} → ((𝐹‘𝐵) = 𝑋 ∨ (𝐹‘𝐵) = 𝑌))
12		eqtr3 2783	. . . . . . . 8 ⊢ (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
13	3, 7	jca 520	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵}))
14		f1veqaeq 7254	. . . . . . . . 9 ⊢ ((𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} ∧ (𝐴 ∈ {𝐴, 𝐵} ∧ 𝐵 ∈ {𝐴, 𝐵})) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
15	13, 14	sylan2 604	. . . . . . . 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 417	. . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑋) → 𝐴 = 𝐵)))
18		eqneqall 2967	. . . . . . . . 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 1151	. . . . . . 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 412	. . . 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 881	. . . . 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 880	. . . . 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 2783	. . . . . . . 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 417	. . . . . 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 412	. . . 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 1052	. . 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 618	. 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 711	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 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  {cpr 4590  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544

This theorem is referenced by: (None)