Theorem resf1ext2b 7931

Description: Extension of an injection which is a restriction of a function. (Contributed by AV, 3-Oct-2025.)

Assertion

Ref	Expression
resf1ext2b	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) ↔ Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))

Proof of Theorem resf1ext2b

Step	Hyp	Ref	Expression
1		fssres 6744	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
2	1	3adant2 1147	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
3		df-f1 6541	. . . . 5 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)))
4		resf1extb 7930	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) ↔ (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵))
5		df-f1 6541	. . . . . . . 8 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵 ∧ Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))
6	5	simprbi 502	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})))
7	4, 6	biimtrdi 256	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))
8	7	expd 420	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})))))
9	3, 8	biimtrrid 246	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)) → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})))))
10	2, 9	mpand 707	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (Fun ◡(𝐹 ↾ 𝐶) → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})))))
11	10	impd 415	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))
12		simp1 1152	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝐹:𝐴⟶𝐵)
13		simp3 1154	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
14		eldifi 4084	. . . . . . 7 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → 𝑋 ∈ 𝐴)
15	14	snssd 4751	. . . . . 6 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → {𝑋} ⊆ 𝐴)
16	15	3ad2ant2 1150	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → {𝑋} ⊆ 𝐴)
17	13, 16	unssd 4144	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐶 ∪ {𝑋}) ⊆ 𝐴)
18	12, 17	fssresd 6745	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵)
19	3	simprbi 502	. . . . . 6 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → Fun ◡(𝐹 ↾ 𝐶))
20	19	anim1i 626	. . . . 5 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) → (Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)))
21	4, 20	biimtrrdi 257	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 → (Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))))
22	5, 21	biimtrrid 246	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵 ∧ Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))) → (Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))))
23	18, 22	mpand 707	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})) → (Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))))
24	11, 23	impbid 215	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) ↔ Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2141   ∉ wnel 3062   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  {csn 4588  ^◡ccnv 5660   ↾ cres 5663   “ cima 5664  Fun wfun 6530  ⟶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-nel 3063  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-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544

This theorem is referenced by:  dfpth2  30044