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Theorem f1mpt 5353
 Description: Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
f1mpt.1 𝐹 = (x A𝐶)
f1mpt.2 (x = y𝐶 = 𝐷)
Assertion
Ref Expression
f1mpt (𝐹:A1-1B ↔ (x A 𝐶 B x A y A (𝐶 = 𝐷x = y)))
Distinct variable groups:   x,y,A   x,B,y   y,𝐶   x,𝐷   y,𝐹
Allowed substitution hints:   𝐶(x)   𝐷(y)   𝐹(x)

Proof of Theorem f1mpt
StepHypRef Expression
1 f1mpt.1 . . . 4 𝐹 = (x A𝐶)
2 nfmpt1 3841 . . . 4 x(x A𝐶)
31, 2nfcxfr 2172 . . 3 x𝐹
4 nfcv 2175 . . 3 y𝐹
53, 4dff13f 5352 . 2 (𝐹:A1-1B ↔ (𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)))
61fmpt 5262 . . 3 (x A 𝐶 B𝐹:AB)
76anbi1i 431 . 2 ((x A 𝐶 B x A y A ((𝐹x) = (𝐹y) → x = y)) ↔ (𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)))
8 f1mpt.2 . . . . . . 7 (x = y𝐶 = 𝐷)
98eleq1d 2103 . . . . . 6 (x = y → (𝐶 B𝐷 B))
109cbvralv 2527 . . . . 5 (x A 𝐶 By A 𝐷 B)
11 raaanv 3322 . . . . . 6 (x A y A (𝐶 B 𝐷 B) ↔ (x A 𝐶 B y A 𝐷 B))
121fvmpt2 5197 . . . . . . . . . . . . . 14 ((x A 𝐶 B) → (𝐹x) = 𝐶)
138, 1fvmptg 5191 . . . . . . . . . . . . . 14 ((y A 𝐷 B) → (𝐹y) = 𝐷)
1412, 13eqeqan12d 2052 . . . . . . . . . . . . 13 (((x A 𝐶 B) (y A 𝐷 B)) → ((𝐹x) = (𝐹y) ↔ 𝐶 = 𝐷))
1514an4s 522 . . . . . . . . . . . 12 (((x A y A) (𝐶 B 𝐷 B)) → ((𝐹x) = (𝐹y) ↔ 𝐶 = 𝐷))
1615imbi1d 220 . . . . . . . . . . 11 (((x A y A) (𝐶 B 𝐷 B)) → (((𝐹x) = (𝐹y) → x = y) ↔ (𝐶 = 𝐷x = y)))
1716ex 108 . . . . . . . . . 10 ((x A y A) → ((𝐶 B 𝐷 B) → (((𝐹x) = (𝐹y) → x = y) ↔ (𝐶 = 𝐷x = y))))
1817ralimdva 2381 . . . . . . . . 9 (x A → (y A (𝐶 B 𝐷 B) → y A (((𝐹x) = (𝐹y) → x = y) ↔ (𝐶 = 𝐷x = y))))
19 ralbi 2439 . . . . . . . . 9 (y A (((𝐹x) = (𝐹y) → x = y) ↔ (𝐶 = 𝐷x = y)) → (y A ((𝐹x) = (𝐹y) → x = y) ↔ y A (𝐶 = 𝐷x = y)))
2018, 19syl6 29 . . . . . . . 8 (x A → (y A (𝐶 B 𝐷 B) → (y A ((𝐹x) = (𝐹y) → x = y) ↔ y A (𝐶 = 𝐷x = y))))
2120ralimia 2376 . . . . . . 7 (x A y A (𝐶 B 𝐷 B) → x A (y A ((𝐹x) = (𝐹y) → x = y) ↔ y A (𝐶 = 𝐷x = y)))
22 ralbi 2439 . . . . . . 7 (x A (y A ((𝐹x) = (𝐹y) → x = y) ↔ y A (𝐶 = 𝐷x = y)) → (x A y A ((𝐹x) = (𝐹y) → x = y) ↔ x A y A (𝐶 = 𝐷x = y)))
2321, 22syl 14 . . . . . 6 (x A y A (𝐶 B 𝐷 B) → (x A y A ((𝐹x) = (𝐹y) → x = y) ↔ x A y A (𝐶 = 𝐷x = y)))
2411, 23sylbir 125 . . . . 5 ((x A 𝐶 B y A 𝐷 B) → (x A y A ((𝐹x) = (𝐹y) → x = y) ↔ x A y A (𝐶 = 𝐷x = y)))
2510, 24sylan2b 271 . . . 4 ((x A 𝐶 B x A 𝐶 B) → (x A y A ((𝐹x) = (𝐹y) → x = y) ↔ x A y A (𝐶 = 𝐷x = y)))
2625anidms 377 . . 3 (x A 𝐶 B → (x A y A ((𝐹x) = (𝐹y) → x = y) ↔ x A y A (𝐶 = 𝐷x = y)))
2726pm5.32i 427 . 2 ((x A 𝐶 B x A y A ((𝐹x) = (𝐹y) → x = y)) ↔ (x A 𝐶 B x A y A (𝐶 = 𝐷x = y)))
285, 7, 273bitr2i 197 1 (𝐹:A1-1B ↔ (x A 𝐶 B x A y A (𝐶 = 𝐷x = y)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ↦ cmpt 3809  ⟶wf 4841  –1-1→wf1 4842  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fv 4853 This theorem is referenced by: (None)
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