Theorem imasetpreimafvbijlemf1 43613

Description: Lemma for imasetpreimafvbij 43615: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)

Hypotheses

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
fundcmpsurinj.h	⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))

Assertion

Ref	Expression
imasetpreimafvbijlemf1	⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃

Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)

Proof of Theorem imasetpreimafvbijlemf1

Dummy variables 𝑎 𝑏 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fundcmpsurinj.p	. . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2		fundcmpsurinj.h	. . 3 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
3	1, 2	imasetpreimafvbijlemf 43610	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	1, 2	imasetpreimafvbijlemfv1 43612	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → ∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏))
5	1, 2	imasetpreimafvbijlemfv1 43612	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑟 ∈ 𝑃) → ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎))
6	4, 5	anim12dan 620	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) ∧ ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎)))
7		eqeq12 2835	. . . . . . . . . . . 12 ⊢ (((𝐻‘𝑠) = (𝐹‘𝑏) ∧ (𝐻‘𝑟) = (𝐹‘𝑎)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
8	7	ancoms 461	. . . . . . . . . . 11 ⊢ (((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏)) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
9	8	adantl 484	. . . . . . . . . 10 ⊢ (((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) ∧ ((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏))) → ((𝐻‘𝑠) = (𝐻‘𝑟) ↔ (𝐹‘𝑏) = (𝐹‘𝑎)))
10		simplll 773	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → 𝐹 Fn 𝐴)
11		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃))
12		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → 𝑏 ∈ 𝑠)
13	12	anim1i 616	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟))
14	1	elsetpreimafveq 43606	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃) ∧ (𝑏 ∈ 𝑠 ∧ 𝑎 ∈ 𝑟)) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
15	10, 11, 13, 14	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
16	15	adantr 483	. . . . . . . . . 10 ⊢ (((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) ∧ ((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏))) → ((𝐹‘𝑏) = (𝐹‘𝑎) → 𝑠 = 𝑟))
17	9, 16	sylbid 242	. . . . . . . . 9 ⊢ (((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) ∧ ((𝐻‘𝑟) = (𝐹‘𝑎) ∧ (𝐻‘𝑠) = (𝐹‘𝑏))) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
18	17	exp32 423	. . . . . . . 8 ⊢ ((((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑎 ∈ 𝑟) → ((𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐹‘𝑏) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
19	18	rexlimdva 3284	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → (∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐹‘𝑏) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
20	19	com23 86	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) ∧ 𝑏 ∈ 𝑠) → ((𝐻‘𝑠) = (𝐹‘𝑏) → (∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
21	20	rexlimdva 3284	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → (∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) → (∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))))
22	21	impd 413	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → ((∃𝑏 ∈ 𝑠 (𝐻‘𝑠) = (𝐹‘𝑏) ∧ ∃𝑎 ∈ 𝑟 (𝐻‘𝑟) = (𝐹‘𝑎)) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟)))
23	6, 22	mpd 15	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑠 ∈ 𝑃 ∧ 𝑟 ∈ 𝑃)) → ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
24	23	ralrimivva 3191	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟))
25		dff13 7013	. 2 ⊢ (𝐻:𝑃–1-1→(𝐹 “ 𝐴) ↔ (𝐻:𝑃⟶(𝐹 “ 𝐴) ∧ ∀𝑠 ∈ 𝑃 ∀𝑟 ∈ 𝑃 ((𝐻‘𝑠) = (𝐻‘𝑟) → 𝑠 = 𝑟)))
26	3, 24, 25	sylanbrc 585	1 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  {csn 4567  ∪ cuni 4838   ↦ cmpt 5146  ^◡ccnv 5554   “ cima 5558   Fn wfn 6350  ⟶wf 6351  –1-1→wf1 6352  ‘cfv 6355

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fv 6363

This theorem is referenced by:  imasetpreimafvbij  43615