Theorem resf1ext2b 7871

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 6695	. . . . 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 6492	. . . . 5 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ Fun ◡(𝐹 ↾ 𝐶)))
4		resf1extb 7870	. . . . . . 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 6492	. . . . . . . 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 4080	. . . . . . 7 ⊢ (𝑋 ∈ (𝐴 ∖ 𝐶) → 𝑋 ∈ 𝐴)
15	14	snssd 4760	. . . . . 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 4141	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ (𝐴 ∖ 𝐶) ∧ 𝐶 ⊆ 𝐴) → (𝐶 ∪ {𝑋}) ⊆ 𝐴)
18	12, 17	fssresd 6696	. . 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 2111   ∉ wnel 3032   ∖ cdif 3894   ∪ cun 3895   ⊆ wss 3897  {csn 4575  ^◡ccnv 5618   ↾ cres 5621   “ cima 5622  Fun wfun 6481  ⟶wf 6483  –1-1→wf1 6484  ‘cfv 6487

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fv 6495

This theorem is referenced by:  dfpth2  29714