Theorem resf1ext2b 7926

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 6741	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
2	1	3adant2 1131	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
3		df-f1 6533	. . . . 5 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)))
4		resf1extb 7925	. . . . . . 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 6533	. . . . . . . 8 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 ↔ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵 ∧ Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))
6	5	simprbi 496	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})–1-1→𝐵 → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})))
7	4, 6	biimtrdi 253	. . . . . 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 415	. . . . 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 243	. . . 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 695	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (Fun ◡(𝐹 ↾ 𝐶) → ((𝐹‘𝑋) ∉ (𝐹 “ 𝐶) → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})))))
11	10	impd 410	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → ((Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) → Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋}))))
12		simp1 1136	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝐹:𝐴⟶𝐵)
13		simp3 1138	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
14		eldifi 4104	. . . . . . 7 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → 𝑋 ∈ 𝐴)
15	14	snssd 4783	. . . . . 6 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → {𝑋} ⊆ 𝐴)
16	15	3ad2ant2 1134	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → {𝑋} ⊆ 𝐴)
17	13, 16	unssd 4165	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐶 ∪ {𝑋}) ⊆ 𝐴)
18	12, 17	fssresd 6742	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ (𝐶 ∪ {𝑋})):(𝐶 ∪ {𝑋})⟶𝐵)
19	3	simprbi 496	. . . . . 6 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → Fun ◡(𝐹 ↾ 𝐶))
20	19	anim1i 615	. . . . 5 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)) → (Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶)))
21	4, 20	biimtrrdi 254	. . . 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 243	. . 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 695	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (Fun ◡(𝐹 ↾ (𝐶 ∪ {𝑋})) → (Fun ◡(𝐹 ↾ 𝐶) ∧ (𝐹‘𝑋) ∉ (𝐹 “ 𝐶))))
24	11, 23	impbid 212	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 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2107   ∉ wnel 3035   ∖ cdif 3921   ∪ cun 3922   ⊆ wss 3924  {csn 4599  ^◡ccnv 5651   ↾ cres 5654   “ cima 5655  Fun wfun 6522  ⟶wf 6524  –1-1→wf1 6525  ‘cfv 6528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fv 6536

This theorem is referenced by:  dfpth2  29645